Sequence 1From Wikipedia, the free encyclopediaContents1 Almost convergent sequence 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Arithmetic progression 22.1 Sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Formulas at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Cauchy product 63.1 Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.1 Cauchy product of two nite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 Cauchy product of two innite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.3 Cauchy product of two nite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.4 Cauchy product of two innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.5 Cauchy product of two power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Convergence and Mertens theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3.2 Proof of Mertens theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4.1 Finite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4.2 Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Cesros theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.6.1 Products of nitely many innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.7 Relation to convolution of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10iii CONTENTS3.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Cauchy sequence 134.1 In real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 In a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3.2 Counter-example: rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3.3 Counter-example: open interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.4 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.1 In topological vector spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.2 In topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.3 In groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.4 In constructive mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.5 In a hyperreal continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Chebyshevs sum inequality 185.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Continuous version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Complementary sequences 206.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 Properties of complementary pairs of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.4 Golay pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.5 Applications of complementary sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Cutting sequence 247.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 DavenportSchinzel sequence 258.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2 Length bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 Application to lower envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26CONTENTS iii8.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Disjunctive sequence 299.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2 Rich numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110Divisibility sequence 3210.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211Ducci sequence 3411.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.3Modulo two form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.4Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.5Other related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612Farey sequence 3712.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812.3.1 Sequence length and index of a fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812.3.2 Farey neighbours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.3.4 Ford circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.3.5 Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4Next term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513Geometric progression 4613.1Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.2Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48iv CONTENTS13.2.2 Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.2.3 Innite geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.2.4 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.3Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.4Relationship to geometry and Euclids work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314Halton sequence 5414.1Example of Halton sequence used to generate points in (0, 1) (0, 1) in R2. . . . . . . . . . . . . 5414.2Implementation in Pseudo Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615Harmonic progression (mathematics) 5715.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.2Use in geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816Innite product 5916.1Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5916.2Product representations of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117Interleave sequence 6217.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218Iterated function 6318.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.2Abelian property and Iteration sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.3Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.4Limiting behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.5Fractional iterates and ows, and negative iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.6Some formulas for fractional iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.6.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6518.6.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66CONTENTS v18.7Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.8Markov chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.9Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.10Means of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.11In computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.12Denitions in terms of iterated functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.13Lies data transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819Katydid sequence 6919.1Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6919.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6919.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920Limit of a sequence 7020.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.2Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7120.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7120.2.2 Formal Denition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7220.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7220.2.4 Innite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7220.3Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.4Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.4.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.5Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.6Denition in hyperreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7320.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.8.1 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7520.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521List of sums of reciprocals 7621.1Finitely many terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7621.2Innitely many terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.2.1 Convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7721.2.2 Divergent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7821.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78vi CONTENTS21.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822Logarithmically concave sequence 7922.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7922.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923Low-discrepancy sequence 8023.1Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8023.1.1 Low-discrepancy sequences in numerical integration . . . . . . . . . . . . . . . . . . . . 8023.2Denition of discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8123.3The KoksmaHlawka inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.4The formula of Hlawka-Zaremba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.5The L2version of the KoksmaHlawka inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 8223.6The ErdsTurnKoksma inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8323.7The main conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8323.8Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8423.9Construction of low-discrepancy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8423.9.1 Random numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8423.9.2 Additive recurrence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8523.9.3 Sobol sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8523.9.4 van der Corput sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.9.5 Halton sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.9.6 Hammersley set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8623.9.7 Poisson disk sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8723.10Graphical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8723.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8723.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8724Mathematics of oscillation 9524.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9524.1.1 Oscillation of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9524.1.2 Oscillation of a function on an open set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.1.3 Oscillation of a function at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.3Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9624.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9824.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9825Monotone convergence theorem 9925.1Convergence of a monotone sequence of real numbers . . . . . . . . . . . . . . . . . . . . . . . . 9925.1.1 Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925.1.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99CONTENTS vii25.1.3 Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925.1.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925.1.5 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925.1.6 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9925.2Convergence of a monotone series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.2.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.3Lebesgues monotone convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.3.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10025.3.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10125.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10325.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10326Periodic sequence 10426.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10426.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10426.3Periodic 0, 1 sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10526.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10527Polynomial sequence 10627.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10627.2Classes of polynomial sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10727.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10728Polyphase sequence 10828.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10828.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10829Random sequence 10929.1Early history. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11029.2Modern approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11029.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11029.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11129.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11129.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11130Sequence 11230.1Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11330.1.1 Important examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11330.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11430.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11530.2Formal denition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11530.2.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115viii CONTENTS30.2.2 Finite and innite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11630.2.3 Increasing and decreasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11630.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11630.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11630.3Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11730.3.1 Denition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11830.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11830.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.4Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11930.5Use in other elds of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12030.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12030.5.2 Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12030.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12130.5.5 Set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.6Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12230.7Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.8Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12330.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12431Sequence space 12531.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12531.1.1 pspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12531.1.2 c and c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12631.1.3 Other sequence spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12631.2Properties of pspaces and the space c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12731.2.1 pspaces are increasing in p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12831.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12831.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12832Shift rule 12932.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12933Sobol sequence 13033.1Good distributions in the s-dimensional unit hypercube . . . . . . . . . . . . . . . . . . . . . . . 13033.2A fast algorithm for the construction of Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . 13133.3Additional uniformity properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13133.4The initialization of Sobol numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132CONTENTS ix33.5Implementation and availability of Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . . . 13233.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13233.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13333.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13333.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13334Stationary sequence 13434.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13434.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435Sturmian word 13535.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13535.1.1 Combinatoric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13535.1.2 Geometric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13635.2Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13635.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13635.2.2 Balanced aperiodic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13735.2.3 Slope and intercept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13735.2.4 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.3Non-binary words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.4Associated real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.5History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13835.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13935.8Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13936Subadditivity 14036.1Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14036.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14036.3Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14136.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14136.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14136.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14136.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14237Subsequence 14337.1Common subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.2Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.3Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14437.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14437.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14438Subsequential limit 145x CONTENTS39Superadditivity 14639.1Examples of superadditive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14639.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14639.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14740Tuple 14840.1Etymology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14840.1.1 Names for tuples of specic lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14840.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14840.3Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14940.3.1 Tuples as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14940.3.2 Tuples as nested ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14940.3.3 Tuples as nested sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15040.4 n-tuples of m-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15040.5Type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15040.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15140.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15140.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15241Van der Corput sequence 15341.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15441.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15442Vites formula 15542.1Signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15642.2Interpretation and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15642.3Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15742.4Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15742.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15842.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15942.7Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 16042.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16042.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16342.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Chapter 1Almost convergent sequenceA bounded real sequence (xn) is said to be almost convergent to L if each Banach limit assigns the same value L tothe sequence (xn) .Lorentz proved that (xn) is almost convergent if and only iflimpxn +. . . +xn+p1p= Luniformly in n .The above limit can be rewritten in detail as( > 0)(p0)(p > p0)(n)xn +. . . +xn+p1pL< .Almost convergence is studied in summability theory. It is an example of a summability method which cannot berepresented as a matrix method.1.1 ReferencesG. Bennett and N.J. Kalton: Consistency theorems for almost convergence. Trans. Amer. Math. Soc.,198:23-43, 1974.J. Boos: Classical and modern methods in summability. Oxford University Press, New York, 2000.J. Connor and K.-G. Grosse-Erdmann: Sequential denitions of continuity for real functions. Rocky Mt. J.Math., 33(1):93-121, 2003.G.G. Lorentz: A contribution to the theory of divergent sequences. Acta Math., 80:167-190, 1948.This article incorporates material fromAlmost convergent on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.1Chapter 2Arithmetic progressionIn mathematics, an arithmetic progression(AP) or arithmetic sequence is a sequence of numbers such that thedierence between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 is an arithmeticprogression with common dierence of 2.If the initial term of an arithmetic progression is a1 and the common dierence of successive members is d, then thenth term of the sequence ( an ) is given by:an= a1 + (n 1)d,and in generalan= am + (n m)d.A nite portion of an arithmetic progression is called a nite arithmetic progression and sometimes just called anarithmetic progression. The sum of a nite arithmetic progression is called an arithmetic series.The behavior of the arithmetic progression depends on the common dierence d. If the common dierence is:Positive, the members (terms) will grow towards positive innity.Negative, the members (terms) will grow towards negative innity.2.1 SumThis section is about Finite arithmetic series. For Innite arithmetic series, see Innite arithmetic series.Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, theresulting sequence has a single repeated value in it, equal to the sum of the rst and last numbers (2 + 14 = 16). Thus16 5 = 80 is twice the sum.The sum of the members of a nite arithmetic progression is called an arithmetic series. For example, consider thesum:2 + 5 + 8 + 11 + 14This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of therst and last number in the progression (here 2 + 14 = 16), and dividing by 2:n(a1 +an)222.2. PRODUCT 3In the case above, this gives the equation:2 + 5 + 8 + 11 + 14 =5(2 + 14)2=5 162= 40.This formula works for any real numbers a1 and an . For example:_32_+_12_+12=3_32+12_2= 32.2.1.1 DerivationTo derive the above formula, begin by expressing the arithmetic series in two dierent ways:Sn= a1 + (a1 +d) + (a1 + 2d) + + (a1 + (n 2)d) + (a1 + (n 1)d)Sn= (an (n 1)d) + (an (n 2)d) + + (an 2d) + (an d) +an.Adding both sides of the two equations, all terms involving d cancel:2Sn= n(a1 +an).Dividing both sides by 2 produces a common form of the equation:Sn=n2(a1 +an).An alternate form results from re-inserting the substitution:an= a1 + (n 1)d :Sn=n2[2a1 + (n 1)d].Furthermore the mean value of the series can be calculated via:Sn/n :n =a1 +an2.In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics andIndian astronomy, gave this method in the Aryabhatiya (section 2.18).2.2 ProductThe product of the members of a nite arithmetic progression with an initial element a1, common dierences d, andn elements in total is determined in a closed expressiona1a2 an= da1dd(a1d+ 1)d(a1d+ 2) d(a1d+n 1) = dn_a1d_n= dn(a1/d +n)(a1/d),where xndenotes the rising factorial and denotes the Gamma function. (Note however that the formula is not validwhen a1/d is a negative integer or zero.)This is a generalization from the fact that the product of the progression 1 2 n is given by the factorial n!and that the product4 CHAPTER 2. ARITHMETIC PROGRESSIONm(m+ 1) (m+ 2) (n 2) (n 1) nfor positive integers m and n is given byn!(m1)!.Taking the example from above, the product of the terms of the arithmetic progression given by an = 3 + (n1)(5)up to the 50th term isP50= 550(3/5 + 50)(3/5) 3.78438 1098.2.3 Standard deviationThe standard deviation of any arithmetic progression can be calculated via:= |d|(n 1)(n + 1)12where n is the number of terms in the progression, and d is the common dierence between terms2.4 IntersectionsThe intersection of any two doubly-innite arithmetic progressions is either empty or another arithmetic progression,which can be found using the Chinese remainder theorem. If each two progressions in a family of doubly-innitearithmetic progressions have a non-empty intersection, then there exists a number common to all of them; that is,innite arithmetic progressions form a Helly family.[1] However, the intersection of innitely many innite arithmeticprogressions might be a single number rather than itself being an innite progression.2.5 Formulas at a GlanceIfa1andnSnnthenan= a1 + (n 1)d,an= am + (n m)d.Sn=n2[2a1 + (n 1)d].Sn=n(a1 +an)22.6. SEE ALSO 55.n = Sn/nn =a1 +an2.2.6 See alsoArithmetico-geometric sequenceGeneralized arithmetic progression - is a set of integers constructed as an arithmetic progression is, but allowingseveral possible dierences.Harmonic progressionHeronian triangles with sides in arithmetic progressionProblems involving arithmetic progressionsUtonality2.7 References[1] Duchet, Pierre (1995), Hypergraphs, in Graham, R. L.; Grtschel, M.; Lovsz, L., Handbook of combinatorics, Vol. 1,2, Amsterdam: Elsevier, pp. 381432, MR 1373663. See in particular Section 2.5, Helly Property, pp. 393394.Sigler, Laurence E. (trans.) (2002). Fibonaccis Liber Abaci. Springer-Verlag. pp. 259260. ISBN 0-387-95419-8.2.8 External linksHazewinkel, Michiel, ed. (2001), Arithmetic series, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Weisstein, Eric W., Arithmetic progression, MathWorld.Weisstein, Eric W., Arithmetic series, MathWorld.Chapter 3Cauchy productIn mathematics, more specically in mathematical analysis, the Cauchy product is the discrete convolution of twosequences or two series. It is named after the French mathematician Augustin Louis Cauchy.3.1 DenitionsThe Cauchy product may apply to nite sequences,[1][2] innite sequences, nite series,[3] innite series,[4][5][6][7][8][9][10][11][12][13][14]power series,[15][16] etc. Convergence issues are discussed further down in the sections on Mertens theorem andCesros theorem.3.1.1 Cauchy product of two nite sequencesLet {ai} and {bj} be two nite sequences of complex numbers with the same length n. The Cauchy product of thesetwo nite sequences is equal to the Cauchy product of the nite seriesni=0ai andnj=0bj .3.1.2 Cauchy product of two innite sequencesLet {ai} and {bj} be two innite sequences of complex numbers. The Cauchy product of these two innite sequencesis equal to the Cauchy product of the innite seriesi=0ai andj=0bj .3.1.3 Cauchy product of two nite seriesLetni=0ai andnj=0bj be two nite series with complex terms. The Cauchy product of these two nite series isdened by a discrete convolution as follows:_ni=0ai___nj=0bj__=nk=0ckwhere ck=kl=0albkl3.1.4 Cauchy product of two innite seriesLeti=0ai andj=0bj be two innite series with complex terms. The Cauchy product of these two innite seriesis dened by a discrete convolution as follows:_ i=0ai___j=0bj__=k=0ckwhere ck=kl=0albkl63.2. PROPERTY 73.1.5 Cauchy product of two power seriesConsider the following two power series with complex coecients {ai} and {bj} :i=0aixiandj=0bjxjThe Cauchy product of these two power series is dened by a discrete convolution as follows:_ i=0aixi___j=0bjxj__=k=0ckxkwhere ck=kl=0albklIf these power series are formal power series, then we are manipulating series in disregard of any question ofconvergence: they need not be convergent series. Otherwise, see Mertens theorem and Cesros theorem belowfor convergence criteria.3.2 PropertyLetni=0ai andnj=0bj be two nite series with complex terms. The product of these two nite series satises theequation:_nk=0ak__nk=0bk_=2nk=0ki=0aibki n1k=0_ak2nki=n+1bi +bk2nki=n+1ai_3.3 Convergence and Mertens theoremNot to be confused with Mertens theorems concerning distribution of prime numbers.Let (an)n and (bn)n be real or complex sequences. It was proved by Franz Mertens that, if the seriesn=0anconverges to A andn=0bn converges to B, and at least one of them converges absolutely, then their Cauchy productconverges to AB.It is not sucient for both series to be convergent; if both sequences are conditionally convergent, the Cauchy productdoes not have to converge towards the product of the two series, as the following example shows:3.3.1 ExampleConsider the two alternating series withan= bn=(1)nn + 1,which are only conditionally convergent (the divergence of the series of the absolute values follows from the directcomparison test and the divergence of the harmonic series). The terms of their Cauchy product are given bycn=nk=0(1)kk + 1(1)nkn k + 1= (1)nnk=01(k + 1)(n k + 1)8 CHAPTER 3. CAUCHY PRODUCTfor every integer n 0. Since for every k {0, 1, ..., n} we have the inequalities k + 1 n + 1 and n k + 1 n +1, it follows for the square root in the denominator that (k + 1)(n k + 1) n +1, hence, because there are n + 1summands,|cn| nk=01n + 1 1for every integer n 0. Therefore, cn does not converge to zero as n , hence the series of the (cn)n divergesby the term test.3.3.2 Proof of Mertens theoremAssume without loss of generality that the seriesn=0an converges absolutely. Dene the partial sumsAn=ni=0ai, Bn=ni=0biand Cn=ni=0ciwithci=ik=0akbik .ThenCn=ni=0aniBiby rearrangement, henceFix > 0. SincekN|ak|< by absolute convergence, and since Bn converges to B as n , there exists aninteger N such that, for all integers n N,(this is the only place where the absolute convergence is used). Since the series of the (an)n converges, the individualan must converge to 0 by the term test. Hence there exists an integer M such that, for all integers n M,Also, since An converges to A as n , there exists an integer L such that, for all integers n L,Then, for all integers n max{L, M + N}, use the representation (1) for Cn, split the sumin two parts, use the triangleinequality for the absolute value, and nally use the three estimates (2), (3) and (4) to show that|Cn AB| =ni=0ani(Bi B) + (An A)BN1i=0|ani..M| |Bi B|. ./(3N)(3) by+ni=N|ani| |Bi B|. ./3(2) by+|An A| |B|. ./3(4) by .By the denition of convergence of a series, Cn AB as required.3.4. EXAMPLES 93.4 Examples3.4.1 Finite seriesSuppose ai= 0 for all i > n and bi= 0 for all i > m. Here the Cauchy product ofan andbn is readily veriedto be (a0 + +an)(b0 + +bm) . Therefore, for nite series (which are nite sums), Cauchy multiplication isdirect multiplication of those series.3.4.2 Innite seriesFor some x, y R , let an= xn/n! and bn= yn/n! . Thencn=ni=0xii!yni(n i)!=1n!ni=0_ni_xiyni=(x +y)nn!by denition and the binomial formula. Since, formally, exp(x) =an and exp(y) =bn , we have shownthat exp(x + y) =cn . Since the limit of the Cauchy product of two absolutely convergent series is equal to theproduct of the limits of those series, we have proven the formula exp(x +y) = exp(x) exp(y) for all x, y R .As a second example, let an= bn= 1 for all n N . Then cn= n + 1 for all n N so the Cauchy productcn= (1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, . . . ) does not converge.3.5 Cesros theoremIn cases where the two sequences are convergent but not absolutely convergent, the Cauchy product is still Cesrosummable. Specically:If (an)n0 , (bn)n0 are real sequences withan A andbn B then1N_Nn=1ni=1ik=0akbik_ AB.This can be generalised to the case where the two sequences are not convergent but just Cesro summable:3.5.1 TheoremFor r> 1 and s> 1 , suppose the sequence (an)n0 is (C, r) summable with sum A and (bn)n0 is (C, s)summable with sum B. Then their Cauchy product is (C, r +s + 1) summable with sum AB.3.6 GeneralizationsAll of the foregoing applies to sequences in C (complex numbers). The Cauchy product can be dened for seriesin the Rnspaces (Euclidean spaces) where multiplication is the inner product. In this case, we have the result that iftwo series converge absolutely then their Cauchy product converges absolutely to the inner product of the limits.3.6.1 Products of nitely many innite seriesLet n N such that n 2 (actually the following is also true for n=1 but the statement becomes trivial in thatcase) and letk1=0a1,k1, . . . ,kn=0an,kn be innite series with complex coecients, from which all except then th one converge absolutely, and the n th one converges. Then the series10 CHAPTER 3. CAUCHY PRODUCTk1=0k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2converges and we have:k1=0k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2=nj=1__kj=0aj,kj__This statement can be proven by induction over n : The case for n=2 is identical to the claim about the Cauchyproduct. This is our induction base.The induction step goes as follows: Let the claimbe true for an n Nsuch that n 2 , and letk1=0a1,k1, . . . ,kn+1=0an+1,kn+1be innite series with complex coecients, from which all except the n+1 th one converge absolutely, and the n+1th one converges. We rst apply the induction hypothesis to the series k1=0|a1,k1|, . . . ,kn=0|an,kn| . Weobtain that the seriesk1=0k1k2=0 kn1kn=0|a1,kna2,kn1kn an,k1k2|converges, and hence, by the triangle inequality and the sandwich criterion, the seriesk1=0k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2converges, and hence the seriesk1=0k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2converges absolutely. Therefore, by the induction hypothesis, by what Mertens proved, and by renaming of variables,we have:n+1j=1__kj=0aj,kj__=__kn+1=0=:akn+1..an+1,kn+1_________k1=0=:bk1 .. k1k2=0 kn1kn=0a1,kna2,kn1kn an,k1k2_______=k1=0k1k2=0an+1,k1k2k2k3=0 knkn+1=0a1,kn+1a2,knkn+1 an,k2k3Therefore, the formula also holds for n + 1 .3.7 Relation to convolution of functionsOne can also dene the Cauchy product of doubly innite sequences, thought of as functions on Z . In this casethe Cauchy product is not always dened: for instance, the Cauchy product of the constant sequence 1 with itself,(. . . , 1, . . . ) is not dened. This doesn't arise for singly innite sequences, as these have only nite sums.One has some pairings, for instance the product of a nite sequence with any sequence, and the product 1 .This is related to duality of Lp spaces.3.8. NOTES 113.8 Notes[1] Dyer & Edmunds 2014, p. 190.[2] Weisstein, Cauchy Product.[3] Oberguggenberger & Ostermann 2011, p. 322.[4] Canuto & Tabacco 2015, p. 20.[5] Bloch 2011, p. 463.[6] Friedman & Kandel 2011, p. 204.[7] Ghorpade & Limaye 2006, p. 416.[8] Hijab 2011, p. 43.[9] Montesinos, Zizler & Zizler 2015, p. 98.[10] Oberguggenberger & Ostermann 2011, p. 322.[11] Pedersen 2015, p. 210.[12] Ponnusamy 2012, p. 200.[13] Pugh 2015, p. 210.[14] Sohrab 2014, p. 73.[15] Canuto & Tabacco 2015, p. 53.[16] Mathonline, Cauchy Product of Power Series.3.9 ReferencesApostol, Tom M. (1974), Mathematical Analysis (2nd ed.), Addison Wesley, p. 204, ISBN 978-0-201-00288-1.Bloch, Ethan D. (2011), The Real Numbers and Real Analysis, Springer.Canuto, Claudio; Tabacco, Anita (2015), Mathematical Analysis II (2nd ed.), Springer.Dyer, R.H.; Edmunds, D.E. (2014), From Real to Complex Analysis, Springer.Friedman, Menahem; Kandel, Abraham (2011), Calculus Light, Springer.Ghorpade, Sudhir R.; Limaye, Balmohan V. (2006), A Course in Calculus and Real Analysis, Springer.Hardy, G. H. (1949), Divergent Series, Oxford University Press, p. 227229.Hijab, Omar (2011), Introduction to Calculus and Classical Analysis (3rd ed.), Springer.Mathonline, Cauchy Product of Power Series.Montesinos, Vicente; Zizler, Peter; Zizler, Vclav (2015), An Introduction to Modern Analysis, Springer.Oberguggenberger, Michael; Ostermann, Alexander (2011), Analysis for Computer Scientists, Springer.Pedersen, Steen (2015), From Calculus to Analysis, Springer.12 CHAPTER 3. CAUCHY PRODUCTPonnusamy, S. (2012), Foundations of Mathematical Analysis, Birkhuser.Pugh, Charles C. (2015), Real Mathematical Analysis (2nd ed.), Springer.Sohrab, Houshang H. (2014), Basic Real Analysis (2nd ed.), Birkhuser.Weisstein, Eric W., Cauchy Product, From MathWorld--A Wolfram Web Resource.Chapter 4Cauchy sequence(a) The plot of a Cauchy sequence (xn), shown in blue, as xn versus n If the space containing the sequence is com-plete, the ultimate destination of this sequence (that is, the limit) exists.(b) A sequence that is not Cauchy. The elements of the sequence fail to get arbitrarily close to each other as thesequence progresses.In mathematics, a Cauchy sequence (French pronunciation:[koi]; English pronunciation: /koi/ KOH-shee), namedafter Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequenceprogresses.[1] More precisely, given any small positive distance, all but a nite number of elements of the sequenceare less than that given distance from each other.The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences areknown to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself, asopposed to the denition of convergence, which uses the limit value as well as the terms. This is often exploitedin algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce aCauchy sequence, consisting of the iterates, thus fullling a logical condition, such as termination.The notions above are not as unfamiliar as they might at rst appear. The customary acceptance of the fact that anyreal number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rationalnumbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x. In somecases it may be dicult to describe x independently of such a limiting process involving rational numbers.Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy lters and Cauchynets.1314 CHAPTER 4. CAUCHY SEQUENCE4.1 In real numbersA sequencex1, x2, x3, . . .of real numbers is called a Cauchy sequence, if for every positive real number , there is a positive integer N suchthat for all natural numbers m, n > N|xm xn| < ,where the vertical bars denote the absolute value. In a similar way one can dene Cauchy sequences of rational orcomplex numbers. Cauchy formulated such a condition by requiring xm xn to be innitesimal for every pair ofinnite m, n.4.2 In a metric spaceTo dene Cauchy sequences in any metric space X, the absolute value |x - x| is replaced by the distance d(x, x)(where d : X X R with some specic properties, see Metric (mathematics)) between x and x.Formally, given a metric space (X, d), a sequencex1, x2, x3, ...is Cauchy, if for every positive real number > 0 there is a positive integer N such that for all positive integers m, n> N, the distanced(x, x) < .Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that thesequence ought to have a limit in X. Nonetheless, such a limit does not always exist within X.4.3 CompletenessA metric space X in which every Cauchy sequence converges to an element of X is called complete.4.3.1 ExamplesThe real numbers are complete under the metric induced by the usual absolute value, and one of the standardconstructions of the real numbers involves Cauchy sequences of rational numbers.A rather dierent type of example is aorded by a metric space X which has the discrete metric (where any twodistinct points are at distance 1 from each other). Any Cauchy sequence of elements of X must be constant beyondsome xed point, and converges to the eventually repeating term.4.3.2 Counter-example: rational numbersThe rational numbers Q are not complete (for the usual distance):There are sequences of rationals that converge (in R) to irrational numbers; these are Cauchy sequences having nolimit in Q. In fact, if a real number x is irrational, then the sequence (xn), whose n-th term is the truncation to ndecimal places of the decimal expansion of x, gives a Cauchy sequence of rational numbers with irrational limit x.Irrational numbers certainly exist, for example:4.3. COMPLETENESS 15The sequence dened by x0=1, xn+1=xn+2xn2consists of rational numbers (1, 3/2, 17/12,...), which isclear from the denition; however it converges to the irrational square root of two, see Babylonian method ofcomputing square root.The sequence xn=Fn/Fn1of ratios of consecutive Fibonacci numbers which, if it converges at all, con-verges to a limit satisfying 2= + 1 , and no rational number has this property. If one considers this as asequence of real numbers, however, it converges to the real number = (1+5)/2 , the Golden ratio, whichis irrational.The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are known to be irrational forany rational value of x0, but each can be dened as the limit of a rational Cauchy sequence, using, for instance,the Maclaurin series.4.3.3 Counter-example: open intervalThe open interval X = (0, 2) in the set of real numbers with an ordinary distance in R is not a complete space: thereis a sequence x = 1/n in it, which is Cauchy (for arbitrarily small distance bound d > 0 all terms x of n > 1/d t inthe (0, d) interval), however does not converge in X its 'limit', number 0, does not belong to the space X.4.3.4 Other propertiesEvery convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number > 0, beyondsome xed point, every term of the sequence is within distance /2 of s, so any two terms of the sequence arewithin distance of each other.Every Cauchy sequence of real (or complex) numbers is bounded (since for some N, all terms of the sequencefrom the N-th onwards are within distance 1 of each other, and if M is the largest absolute value of the termsup to and including the N-th, then no term of the sequence has absolute value greater than M+1).In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent(with the same limit), since, given any real number r > 0, beyond some xed point in the original sequence,every term of the subsequence is within distance r/2 of s, and any two terms of the original sequence are withindistance r/2 of each other, so every term of the original sequence is within distance r of s.These last two properties, together with a lemma used in the proof of the BolzanoWeierstrass theorem, yield onestandard proof of the completeness of the real numbers, closely related to both the BolzanoWeierstrass theoremand the HeineBorel theorem. The lemma in question states that every bounded sequence of real numbers has aconvergent monotonic subsequence. Given this fact, every Cauchy sequence of real numbers is bounded, hence has aconvergent subsequence, hence is itself convergent. It should be noted, though, that this proof of the completeness ofthe real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, ofconstructing the real numbers as the completion of the rational numbers, makes the completeness of the real numberstautological.One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use ofcompleteness is provided by consideration of the summation of an innite series of real numbers (or, more generally,of elements of any complete normed linear space, or Banach space). Such a series n=1xn is considered to beconvergent if and only if the sequence of partial sums (sm) is convergent, where sm=mn=1xn . It is a routinematter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers p > q,sp sq=pn=q+1xn.If f :M N is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequencein M, then (f(xn)) is a Cauchy sequence in N. If (xn) and (yn) are two Cauchy sequences in the rational, real orcomplex numbers, then the sum (xn +yn) and the product (xnyn) are also Cauchy sequences.16 CHAPTER 4. CAUCHY SEQUENCE4.4 Generalizations4.4.1 In topological vector spacesThere is also a concept of Cauchy sequence for a topological vector spaceX: Pick a local baseB forXabout0; then (xk) is a Cauchy sequence if for each memberV B , there is some numberNsuch that whenevern, m > N, xn xm is an element of V. If the topology of X is compatible with a translation-invariant metric d ,the two denitions agree.4.4.2 In topological groupsSince the topological vector space denition of Cauchy sequence requires only that there be a continuous subtractionoperation, it can just as well be stated in the context of a topological group: A sequence (xk) in a topological groupG is a Cauchy sequence if for every open neighbourhood U of the identity in G there exists some number N suchthat whenever m, n > N it follows that xnx1m U . As above, it is sucient to check this for the neighbourhoodsin any local base of the identity in G .As in the construction of the completion of a metric space, one can furthermore dene the binary relation on Cauchysequences in G that (xk) and (yk) are equivalent if for every open neighbourhood U of the identity in G there existssome number N such that whenever m, n > N it follows that xny1m U . This relation is an equivalence relation:It is reexive since the sequences are Cauchy sequences. It is symmetric sinceynx1m=(xmy1n)1U1which by continuity of the inverse is another open neighbourhood of the identity. It is transitive sincexnz1l=xny1mymz1l UU where U and U are open neighbourhoods of the identity such that UU U ; such pairsexist by the continuity of the group operation.4.4.3 In groupsThere is also a concept of Cauchy sequence in a groupG : LetH=(Hr) be a decreasing sequence of normalsubgroups of G of nite index. Then a sequence (xn) in G is said to be Cauchy (w.r.t. H ) if and only if for any rthere is N such that m, n > N, xnx1m Hr .Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on G ,namely that for which H is a local base.The set C of such Cauchy sequences forms a group (for the componentwise product), and the set C0 of null sequences(s.th. r, N, n > N, xn Hr ) is a normal subgroup of C . The factor group C/C0 is called the completion ofG with respect to H .One can then show that this completion is isomorphic to the inverse limit of the sequence (G/Hr) .An example of this construction, familiar in number theory and algebraic geometry is the construction of the p-adiccompletion of the integers with respect to a prime p. In this case, G is the integers under addition, and Hr is theadditive subgroup consisting of integer multiples of pr.IfHis a conal sequence (i.e., any normal subgroup of nite index contains someHr), then this completion iscanonical in the sense that it is isomorphic to the inverse limit of (G/H)H , where H varies over all normal subgroupsof nite index. For further details, see ch. I.10 in Lang's Algebra.4.4.4 In constructive mathematicsIn constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to beuseful. If (x1, x2, x3, ...) is a Cauchy sequence in the set X , then a modulus of Cauchy convergence for the sequenceis a function from the set of natural numbers to itself, such that km, n > (k), |xm xn| < 1/k .Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The converse (that every Cauchysequence has a modulus) follows from the well-ordering property of the natural numbers (let (k) be the smallestpossible Nin the denition of Cauchy sequence, taking r to be 1/k ). However, this well-ordering property doesnot hold in constructive mathematics (it is equivalent to the principle of excluded middle). On the other hand, thisconverse also follows (directly) from the principle of dependent choice (in fact, it will follow from the weaker AC00),4.5. SEE ALSO 17which is generally accepted by constructive mathematicians. Thus, moduli of Cauchy convergence are needed directlyonly by constructive mathematicians who (like Fred Richman) do not wish to use any form of choice.That said, using a modulus of Cauchy convergence can simplify both denitions and theorems in constructive analysis.Perhaps even more useful are regular Cauchy sequences, sequences with a given modulus of Cauchy convergence(usually (k)=k or (k)=2k). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent (inthe sense used to form the completion of a metric space) to a regular Cauchy sequence; this can be proved withoutusing any form of the axiom of choice. Regular Cauchy sequences were used by Errett Bishop in his Foundationsof Constructive Analysis, but they have also been used by Douglas Bridges in a non-constructive textbook (ISBN978-0-387-98239-7). However, Bridges also works on mathematical constructivism; the concept has not spread faroutside of that milieu.4.4.5 In a hyperreal continuumA real sequence un: n N has a natural hyperreal extension, dened for hypernatural values H of the index n inaddition to the usual natural n. The sequence is Cauchy if and only if for every innite H and K, the values uH anduK are innitely close, or adequal, i.e.st(uH uK) = 0where st is the standard part function.4.5 See alsoModes of convergence (annotated index)4.6 References[1] Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl0848.130014.7 Further readingBourbaki, Nicolas (1972). Commutative Algebra (English translation ed.). Addison-Wesley. ISBN 0-201-00644-8.Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN978-0-201-55540-0, Zbl 0848.13001Spivak, Michael (1994). Calculus (3rd ed.). Berkeley, CA: Publish or Perish. ISBN 0-914098-89-6.Troelstra, A. S.; D. van Dalen. Constructivism in Mathematics: An Introduction. (for uses in constructivemathematics)4.8 External linksHazewinkel, Michiel, ed. (2001), Fundamental sequence, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4Chapter 5Chebyshevs sum inequalityFor the similarly named inequality in probability theory, see Chebyshevs inequality.In mathematics, Chebyshevs sum inequality, named after Pafnuty Chebyshev, states that ifa1 a2 anandb1 b2 bn,then1nnk=1ak bk _1nnk=1ak__1nnk=1bk_.Similarly, ifa1 a2 anandb1 b2 bn,then1nnk=1akbk _1nnk=1ak_ _1nnk=1bk_. [1]5.1 ProofConsider the sumS=nj=1nk=1(aj ak)(bj bk).185.2. CONTINUOUS VERSION 19The two sequences are non-increasing, therefore aj ak and bj bk have the same sign for any j, k. Hence S 0.Opening the brackets, we deduce:0 2nnj=1ajbj 2nj=1ajnk=1bk,whence1nnj=1ajbj __1nnj=1aj____1nnj=1bk__.An alternative proof is simply obtained with the rearrangement inequality.5.2 Continuous versionThere is also a continuous version of Chebyshevs sum inequality:If f and g are real-valued, integrable functions over [0,1], both non-increasing or both non-decreasing, then10f(x)g(x)dx 10f(x)dx10g(x)dx,with the inequality reversed if one is non-increasing and the other is non-decreasing.5.3 Notes[1] Hardy, G. H.; Littlewood, J. E.; Plya, G. (1988). Inequalities. Cambridge Mathematical Library. Cambridge: CambridgeUniversity Press. ISBN 0-521-35880-9. MR 0944909.Chapter 6Complementary sequencesFor complementary sequences in biology, see complementarity (molecular biology).In applied mathematics, complementary sequences (CS) are pairs of sequences with the useful property that theirout-of-phase aperiodic autocorrelation coecients sum to zero. Binary complementary sequences were rst intro-duced by Marcel J. E. Golay in 1949. In 19611962 Golay gave several methods for constructing sequences of length2Nand gave examples of complementary sequences of lengths 10 and 26. In 1974 R. J. Turyn gave a method forconstructing sequences of length mn from sequences of lengths m and n which allows the construction of sequencesof any length of the form 2N10K26M.Later the theory of complementary sequences was generalized by other authors to polyphase complementary se-quences, multilevel complementary sequences, and arbitrary complex complementary sequences. Complementarysets have also been considered; these can contain more than two sequences.6.1 DenitionLet (a0, a1, ..., aN ) and (b0, b1, ..., bN ) be a pair of bipolar sequences, meaning that a(k) and b(k) have values+1 or 1. Let the aperiodic autocorrelation function of the sequence x be dened byRx(k) =Nk1j=0xjxj+k.Then the pair of sequences a and b is complementary if:Ra(k) +Rb(k) = 0,for k = 1, ..., N 1.Or using Kronecker delta we can write:Ra(k) +Rb(k) = C(k),where C is a constant.So we can say that the sum of autocorrelation functions of complementary sequences is a delta function, which is anideal autocorrelation for many applications like radar pulse compression and spread spectrum telecommunications.6.2 ExamplesAs the simplest example we have sequences of length 2: (+1, +1) and (+1, 1). Their autocorrelation functionsare (2, 1) and (2, 1), which add up to (4, 0).206.3. PROPERTIES OF COMPLEMENTARY PAIRS OF SEQUENCES 21As the next example (sequences of length 4), we have (+1, +1, +1, 1) and (+1, +1, 1, +1). Their autocorre-lation functions are (4, 1, 0, 1) and (4, 1, 0, 1), which add up to (8, 0, 0, 0).One example of length 8 is (+1, +1, +1, 1, +1, +1, 1, +1) and (+1, +1, +1, 1, 1, 1, +1, 1). Theirautocorrelation functions are (8, 1, 0, 3, 0, 1, 0, 1) and (8, 1, 0, 3, 0, 1, 0, 1).An example of length 10 given by Golay is (+1, +1, 1, +1, 1, +1, 1, 1, +1, +1) and (+1, +1, 1, +1, +1,+1, +1, +1, 1, 1). Their autocorrelation functions are (10, 3, 0, 1, 0, 1,2, 1, 2, 1) and (10, 3, 0, 1, 0,1, 2, 1, 2, 1).6.3 Properties of complementary pairs of sequencesComplementary sequences have complementary spectra. As the autocorrelation function and the power spectraform a Fourier pair, complementary sequences also have complementary spectra. But as the Fourier transformof a delta function is a constant, we can writeSa +Sb= CS,where CS is a constant.Sa and Sb are dened as a squared magnitude of the Fourier transform of the sequences. The Fouriertransform can be a direct DFT of the sequences, it can be a DFT of zero padded sequences or it can bea continuous Fourier transform of the sequences which is equivalent to the Z transform for Z = ej.CS spectra is upper bounded. As Sa and Sb are non-negative values we can writeSa= CS Sb< CS,alsoSb< CS.If either of the sequences of the CS pair is inverted (multiplied by 1) they remain complementary. Moregenerally if any of the sequences is multiplied by ej they remain complementary;If either of the sequences is reversed they remain complementary;If either of the sequences is delayed they remain complementary;If the sequences are interchanged they remain complementary;If both sequences are multiplied by the same constant (real or complex) they remain complementary;If both sequences are decimated in time by K they remain complementary. More precisely if from a com-plementary pair (a(k), b(k)) we form a new pair (a(Nk), b(Nk)) with skipped samples discarded then the newsequences are complementary.If alternating bits of both sequences are inverted they remain complementary. In general for arbitrary complexsequences if both sequences are multiplied by ejkn/N(where k is a constant and n is the time index) they remaincomplementary;A new pair of complementary sequences can be formed as [a b] and [a b] where [..] denotes concatenationand a and b are a pair of CS;A new pair of sequences can be formed as {a b} and {a b} where {..} denotes interleaving of sequences.A new pair of sequences can be formed as a + b and a b.22 CHAPTER 6. COMPLEMENTARY SEQUENCES6.4 Golay pairA complementary pair a, b may be encoded as polynomials A(z) = a(0) + a(1)z + ... + a(N 1)zN1 and similarlyfor B(z). The complementarity property of the sequences is equivalent to the condition|A(z)|2+|B(z)|2= 2Nfor all z on the unit circle, that is, |z| = 1. If so, A and B form a Golay pair of polynomials. Examples include theShapiro polynomials, which give rise to complementary sequences of length a power of 2.6.5 Applications of complementary sequencesMultislit spectrometryUltrasound measurementsAcoustic measurementsradar pulse compressionWi-Fi networks,3G CDMA wireless networksOFDM communication systemsTrain wheel detection systems[1][2]Non-destructive tests (NDT)Communicationscoded aperture masks are designed using a 2-dimensional generalization of complementary sequences.6.6 See alsoPseudorandom binary sequences (also called maximum length sequences or M-sequences)Gold sequencesKasami sequencesPolyphase sequenceWalshHadamard sequencesZadoChu sequenceBinary Golay code (Error-correcting code)Ternary Golay code (Error-correcting code)6.7. REFERENCES 236.7 ReferencesGolay, Marcel J.E. (1949). Multislit spectroscopy. J. Opt. Soc. Am. 39: 437444. doi:10.1364/JOSA.39.000437.Golay, MarcelJ.E. (April1961). Complementaryseries. IRETrans. Inform. Theory7(2): 8287.doi:10.1109/TIT.1961.1057620.Golay, Marcel J.E. (1962). Note on Complementary series". Proc. IRE50: 84. doi:10.1109/JRPROC.1962.288278.Turyn, R.J. (1974). Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression, andsurface wave encodings. J. Combin. Theory (A) 16 (3): 313333. doi:10.1016/0097-3165(74)90056-9.Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. Springer. pp. 1109.ISBN 978-0-387-95444-8.Donato, P.G.; Urea, J.; Mazo, M.; De Marziani, C.; Ochoa, A. (2006). Design and signal processing ofa magnetic sensor array for train wheel detection. Sensors and Actuators A: Physical 132 (2): 516525.doi:10.1016/j.sna.2006.02.043.[1] Donato, P.G.; Urea, J.; Mazo, M.; Alvarez, F. Train wheel detection without electronic equipment near the rail line.2004. doi:10.1109/IVS.2004.1336500[2] J.J. Garcia; A. Hernandez; J. Urea; J.C. Garcia; M. Mazo; J.L. Lazaro; M.C. Perez; F. Alvarez. Low cost obstacledetection for smart railway infrastructures. 2004.Chapter 7Cutting sequenceThe Fibonacci word is an example of a Sturmian word. The start of the cutting sequence shown here illustrates the start of the word0100101001.In digital geometry, a cutting sequence is a sequence of symbols whose elements correspond to the individual gridlines crossed (cut) as a curve crosses a square grid.[1]Sturmian words are a special case of cutting sequences where the curves are straight lines of irrational slope.[2]7.1 References[1] Monteil, T. (2011). The complexity of tangent words. Electronic Proceedings in Theoretical Computer Science 63: 152.doi:10.4204/EPTCS.63.21.[2] Pytheas Fogg (2002) p.152Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathemat-ics 1794. Editors Berth, Valrie; Ferenczi, Sbastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag.ISBN 3-540-44141-7. Zbl 1014.11015.24Chapter 8DavenportSchinzel sequenceIn combinatorics, a DavenportSchinzel sequence is a sequence of symbols in which the number of times any twosymbols may appear in alternation is limited. The maximum possible length of a DavenportSchinzel sequence isbounded by the number of its distinct symbols multiplied by a small but nonconstant factor that depends on the numberof alternations that are allowed. DavenportSchinzel sequences were rst dened in 1965 by Harold Davenport andAndrzej Schinzel to analyze linear dierential equations. Following Atallah (1985) these sequences and their lengthbounds have also become a standard tool in discrete geometry and in the analysis of geometric algorithms.[1]8.1 DenitionA nite sequence U = u1, u2, u3, is said to be a DavenportSchinzel sequence of order s if it satises the followingtwo properties:1. No two consecutive values in the sequence are equal to each other.2. If x and y are two distinct values occurring in the sequence, then the sequence does not contain a subsequence... x, ... y, ..., x, ..., y, ... consisting of s + 2 values alternating between x and y.For instance, the sequence1, 2, 1, 3, 1, 3, 2, 4, 5, 4, 5, 2, 3is a DavenportSchinzel sequence of order 3: it contains alternating subsequences of length four, such as ...1, ... 2,... 1, ... 2, ... (which appears in four dierent ways as a subsequence of the whole sequence) but it does not containany alternating subsequences of length ve.If a DavenportSchinzel sequence of order s includes n distinct values, it is called an (n,s) DavenportSchinzel se-quence, or a DS(n,s)-sequence.[2]8.2 Length boundsThe complexity of DS(n,s)-sequence has been analyzed asymptotically in the limit as n goes to innity, with theassumption that s is a xed constant, and nearly tight bounds are known for all s. Let s(n) denote the length of thelongest DS(n,s)-sequence. The best bounds known on s involve the inverse Ackermann function(n) = min { m | A(m,m) n },where A is the Ackermann function. Due to the very rapid growth of the Ackermann function, its inverse growsvery slowly, and is at most four for problems of any practical size.[3]Using big O and big notation, the following bounds are known:2526 CHAPTER 8. DAVENPORTSCHINZEL SEQUENCE0(n) = 1.1(n) = n.[4]2(n) = 2n 1.[4] 2n(n) O(n) 3(n) 2n(n) + O(n(n)) .[5] This complexity bound can be realized to within aconstant factor by line segments: there exist arrangements of n line segments in the plane whose lower envelopeshave complexity (n (n)).[6]For even values of s 4,[7]s(n) = n 21t!(n)t(1+o(1)), where t = (s 2)/2.For odd values of s 5 the best known upper bound is [7]s(n) < n 21t!(n)tlog (n)(1+o(1)), where t = (s 3)/2.However, this bound is not known to be tight.[7]The value of s(n) is also known when s is variable but n is a small constant:[8]s(1) = 1s(2) = s + 1s(3) = 3s 2 + (s mod 2)s(4) = 6s 2 + (s mod 2).8.3 Application to lower envelopesThe lower envelope of a set of functions i(x) of a real variable x is the function given by their pointwise minimum:(x) = minii(x).Suppose that these functions are particularly well behaved: they are all continuous, and any two of them are equalon at most s values. With these assumptions, the real line can be partitioned into nitely many intervals withinwhich one function has values smaller than all of the other functions. The sequence of these intervals, labeled bythe minimizing function within each interval, forms a DavenportSchinzel sequence of order s. Thus, any upperbound on the complexity of a DavenportSchinzel sequence of this order also bounds the number of intervals in thisrepresentation of the lower envelope.In the original application of Davenport and Schinzel, the functions under consideration were a set of dierent solu-tions to the same homogeneous linear dierential equation of order s. Any two distinct solutions can have at most svalues in common, so the lower envelope of a set of n distinct solutions forms a DS(n,s)-sequence.The same concept of a lower envelope can also be applied to functions that are only piecewise continuous or thatare dened only over intervals of the real line; however, in this case, the points of discontinuity of the functions andthe endpoints of the interval within which each function is dened add to the order of the sequence. For instance, anon-vertical line segment in the plane can be interpreted as the graph of a function mapping an interval of x valuesto their corresponding y values, and the lower envelope of a collection of line segments forms a DavenportSchinzelsequence of order three because any two line segments can form an alternating subsequence with length at most four.8.4 See alsoSquarefree word8.5. NOTES 2711 1 122 2 233 3 3 44455 5A DavenportSchinzel sequence of order 3 formed by the lower envelope of line segments.8.5 Notes[1] Sharir & Agarwal (1995), pp. x and 2.[2] See Sharir & Agarwal (1995), p. 1, for this notation.[3] Sharir & Agarwal (1995), p.14.[4] Sharir & Agarwal (1995), p.6.[5] Sharir & Agarwal (1995), Chapter 2, pp. 1242; Hart & Sharir (1986); Wiernik & Sharir (1988); Komjth (1988); Klazar(1999); Nivasch (2009).[6] Sharir & Agarwal (1995), Chapter 4, pp. 86114; Wiernik & Sharir (1988).[7] Sharir & Agarwal (1995), Chapter 3, pp. 4385; Agarwal & Sharir (1989); Nivasch (1999).[8] Roselle & Stanton (1970/71).8.6 ReferencesAgarwal, P. K.; Sharir, Micha; Shor, P. (1989), Sharp upper and lower bounds on the length of generalDavenportSchinzel sequences, Journal of Combinatorial Theory, Series A52 (2): 228274, doi:10.1016/0097-3165(89)90032-0, MR 1022320.28 CHAPTER 8. DAVENPORTSCHINZEL SEQUENCEAtallah, Mikhail J. (1985), Some dynamic computational geometry problems, Computers and Mathematicswith Applications 11: 11711181, doi:10.1016/0898-1221(85)90105-1, MR 0822083.Davenport, H.; Schinzel, Andrzej (1965), A combinatorial problem connected with dierential equations,American Journal of Mathematics (The Johns Hopkins University Press) 87 (3): 684694, doi:10.2307/2373068,JSTOR 2373068, MR 0190010.Hart, S.; Sharir, Micha (1986), Nonlinearity of DavenportSchinzel sequences and of generalized path com-pression schemes, Combinatorica 6 (2): 151177, doi:10.1007/BF02579170, MR 0875839.Klazar, M. (1999), On the maximumlengths of DavenportSchinzel sequences, Contemporary Trends in Dis-crete Mathematics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 49, AmericanMathematical Society, pp. 169178.Komjth, Pter (1988), A simplied construction of nonlinear DavenportSchinzel sequences, Journal ofCombinatorial Theory, Series A 49 (2): 262267, doi:10.1016/0097-3165(88)90055-6, MR 0964387.Mullin, R. C.; Stanton, R. G. (1972), A map-theoretic approach to Davenport-Schinzel sequences., PacicJournal of Mathematics 40: 167172, doi:10.2140/pjm.1972.40.167, MR 0302601.Nivasch, Gabriel (2009), Improved bounds and new techniques for DavenportSchinzel sequences and theirgeneralizations, Proc. 20th ACM-SIAM Symp. Discrete Algorithms, pp. 110, arXiv:0807.0484.Roselle, D. P.; Stanton, R. G. (1970/71), Some properties of Davenport-Schinzel sequences, Acta Arithmetica17: 355362, MR 0284414 Check date values in: |date= (help).Sharir, Micha; Agarwal, Pankaj K. (1995), DavenportSchinzel Sequences and Their Geometric Applications,Cambridge University Press, ISBN 0-521-47025-0.Stanton, R. G.; Dirksen, P. H. (1976), Davenport-Schinzel sequences., Ars Combinatoria 1 (1): 4351, MR0409347.Stanton, R. G.; Roselle, D. P. (1970), A result on Davenport-Schinzel sequences, Combinatorial theory andits applications, III (Proc. Colloq., Balatonfred, 1969), Amsterdam: North-Holland, pp. 10231027, MR0304189.Wiernik, Ady; Sharir, Micha (1988), Planar realizations of nonlinear DavenportSchinzel sequences by seg-ments, Discrete & Computational Geometry 3 (1): 1547, doi:10.1007/BF02187894, MR 0918177.8.7 External linksDavenport-Schinzel Sequence, from MathWorld.Davenport-Schinzel Sequences, a section in the book Motion Planning, by Steven M. LaValle.Chapter 9Disjunctive sequenceAdisjunctivesequence is an innite sequence (over a nite alphabet of characters) in which every nite stringappears as a substring. For instance, the binary Champernowne sequence0 1 00 01 10 11 000 001 . . .formed by concatenating all binary strings in shortlex order, clearly contains all the binary strings and so is disjunctive.(The spaces above are not signicant and are present solely to make clear the boundaries between strings). Thecomplexity function of a disjunctive sequence S over an alphabet of size k is pS(n) = kn.[1]Any normal sequence (a sequence in which each string of equal length appears with equal frequency) is disjunctive,but the converse is not true. For example, letting 0ndenote the string of length n consisting of all 0s, consider thesequence0 011 0200 0401 0810 01611 032000 064. . .obtained by splicing exponentially long strings of 0s into the shortlex ordering of all binary strings. Most of thissequence consists of long runs of 0s, and so it is not normal, but it is still disjunctive.9.1 ExamplesThe following result[2][3] can be used to generate a variety of disjunctive sequences:If a1, a2, a3, ..., is a strictly increasing innite sequence of positive integers such that lim n (an /an) = 1,then for any positive integer m and any integer base b 2, there is an an whose expression in base bstarts with the expression of m in base b.(Consequently, the innite sequence obtained by concatenating the base-b expressions for a1, a2, a3, ...,is disjunctive over the alphabet {0, 1, ..., b1}.)Two simple cases illustrate this result:an = nk, where k is a xed positive integer. (In this case, lim n (an / an) = lim n ( (n+1)k/ nk) =lim n (1 + 1/n)k= 1.)E.g., using base-ten expressions, the sequences123456789101112... (k = 1, positive natural numbers),1491625364964... (k = 2, squares),182764125216343... (k = 3, cubes),etc.,2930 CHAPTER 9. DISJUNCTIVE SEQUENCEare disjunctive on {0,1,2,3,4,5,6,7,8,9}.an = pn, where pn is the nth prime number. (In this case, lim n (an / an) = 1 is a consequence of pn ~n ln n.)E.g., the sequences23571113171923... (using base ten),10111011111011110110001 ... (using base two),etc.,are disjunctive on the respective digit sets.Another result[4] that provides a variety of disjunctive sequences is as follows:If an = oor(f(n)), where f is any non-constant polynomial with real coecients such that f(x) > 0 forall x > 0,then the concatenation a1a2a3... (with the an expressed in base b) is a normal sequence in base b, andis therefore disjunctive on {0, 1, ..., b1}.E.g., using base-ten expressions, the sequences818429218031851879211521610... (with f(x) = 2x3- 5x2+ 11x )591215182124273034... (with f(x) = x + e)are disjunctive on {0,1,2,3,4,5,6,7,8,9}.9.2 Rich numbersA rich number or disjunctive number is a real number whose expansion with respect to some base b is a disjunctivesequence over the alphabet {0,...,b1}. Every normal number in base b is disjunctive but not conversely. The realnumber x is rich in base b if and only if the set { x bnmod 1} is dense in the unit interval.[5]A number that is disjunctive to every base is called absolutely disjunctive or is said to be a lexicon. Every string inevery alphabet occurs within a lexicon. A set is called "comeager" or residual if it contains the intersection of acountable family of open dense sets. The set of absolutely disjunctive reals is residual.[6] It is conjectured that everyreal irrational algebraic number is absolutely disjunctive.[7]9.3 Notes[1] Bugeaud (2012) p.91[2] Calude, C.; Priese, L.; Staiger, L. (1997), Disjunctive sequences: An overview, University of Auckland, New Zealand, pp.135[3] Istrate, G.; Pun, Gh. (1994), Some combinatorial properties of self-reading sequences, Discrete Applied Mathematics55: 8386, doi:10.1016/0166-218X(94)90037-X, Zbl 0941.68656[4] http://matwbn.icm.edu.pl/ksiazki/aa/aa81/aa8143.pdf[5] Bugeaud (2012) p.92[6] Calude & Zamrescu (1999)[7] Adamczewski & Bugeaud (2010) p.4149.4. REFERENCES 319.4 ReferencesAdamczewski, Boris; Bugeaud, Yann (2010). 8. Transcendence and diophantine approximation. In Berth,Valrie; Rigo, Michael. Combinatorics, automata, and number theory. Encyclopedia of Mathematics and itsApplications 135. Cambridge: Cambridge University Press. p. 410451. ISBN 978-0-521-51597-9. Zblpre05879512.Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Math-ematics 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl pre06066616.Calude, C.S.; Zamrescu, T. (1999). Most numbers obey no probability laws. Publicationes MathematicaeDebrecen 54 (Supplement): 619623.Chapter 10Divisibility sequenceIn mathematics, a divisibility sequence is an integer sequence (an)nN such that for all natural numbers m, n,ifm | n then am | an,i.e., whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term.The concept can be generalized to sequences with values in any ring where the concept of divisibility is dened.A strong divisibility sequence is an integer sequence (an)nN such that for all natural numbers m, n,gcd(am, an) = agcd(m,n).Note that a strong divisibility sequence is immediately a divisibility sequence; if m | n , immediately gcd(m, n) = m. Then by the strong divisibility property, gcd(am, an) = am and therefore am | an .10.1 ExamplesAny constant sequence is a strong divisibility sequence.Every sequence of the form an= kn , for some nonzero integer k, is a divisibility sequence.Every sequence of the form an= AnBnfor integers A > B> 0 is a divisibility sequence.The Fibonacci numbers F = (1, 1, 2, 3, 5, 8,...)