Copyright © 2011 Pearson Education, Inc. Slide 11.1-1

11.1 Sequences

A sequence is a function that has a set of natural numbers (positive integers) as its domain.

Copyright © 2011 Pearson Education, Inc. Slide 11.1-2

11.1 Sequences

A sequence is often specified by giving a formula forthe general term or nth term, an.

Example Find the first four terms for the sequence

Copyright © 2011 Pearson Education, Inc. Slide 11.1-3

11.1 Sequences

• A finite sequence has domain the finite set

{1, 2, 3, …, n} for some natural number n.

Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

• An infinite sequence has domain

{1, 2, 3, …}, the set of all natural numbers.

Example 1, 2, 4, 8, 16, 32, …

Copyright © 2011 Pearson Education, Inc. Slide 11.1-4

11.1 Convergent and Divergent Sequences

• A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number.

• A sequence that is not convergent is said to be divergent.

Copyright © 2011 Pearson Education, Inc. Slide 11.1-5

11.1 Sequences and Recursion Formulas

• A recursion formula or recursive definition defines a sequence by– Specifying the first few terms of the sequence

– Using a formula to specify subsequent terms in terms of preceding terms.

Copyright © 2011 Pearson Education, Inc. Slide 11.1-6

11.1 Series and Summation Notation

• is the sum of the first n terms of the

sequence.

= a1+ a2+ a3+ … + an1

n

ii

a

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Summation Properties

(a)

(b)

(c)

1

n

i

c nc

1 1

n n

i ii i

ca c a

1 1 1

( )n n n

i i i ii i i

a b a b

Copyright © 2011 Pearson Education, Inc. Slide 11.1-8

11.1 Series and Summation NotationA finite series is the sum of the first n terms of the sequence:

Infinite series is the sum of all the terms of the infinite sequence:

.

1 2 31

...n

n n ii

S a a a a a

1 2 31

... ...n ii

S a a a a a