Ad calculus 5
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Transcript of Ad calculus 5
Lecture - 5
1
Test for Convergence
Contents
•Series of Positive terms
•Algebra of sums
•Comparison Test
2
•Let (an) = a1, a2,……an… be a
sequence of real numbers. (ai > 0)
•Notion of sequence of partial sums
follows here too
However, s1= a1
s2 = a1 + a2 ………
Note s1 < s2 < s3 < ……< sn < ……
3
Series of Positive terms
•Hence, we have a monotonically
increasing sequence
•If (sn) is bounded, or unbounded then
will be a convergent series or a
divergent series
•A series of positive term will never
oscillate
4
Series of Positive terms
a1n
n
•If converges to a and
converges to b then
• converges to a + b
• converges to ka
5
Infinite Series – Algebra of sums
a1n
n
b1n
n
)b(a1n
nn
ka1n
n
convergence
Infinite Series of positive terms
Usage of Geometric Series
Algebra of sums
Comparison Test
6
Summary
1. Prove that (an) is convergent if
and only if is
convergent
2. Prove that sum of a convergent
and a divergent series will diverge
3. Test the convergence of
7
Questions
n
1
1n
)a - (a1n
n1n
n
1
1n2