9.2Series and Convergence

If we add all the terms of a sequence, we get a series:

1 2 31

n kk

a a a a a

a1, a2,… are terms of the series. an is the nth term.

To find the sum of a series, we need to consider the partial sums:

1 1S a

2 1 2S a a

3 1 2 3S a a a

1

n

n kk

S a

nth partial sum

If Sn has a limit as , then the series converges,

otherwise it diverges.

n

Series

Examples

1n

n

0

)1(n

n

2

22 1n

n

n

Determine whether the series is convergent or divergent.

Divergence Test

If then the series diverges.lim 0nn a 1

nn

a

Examples: Determine whether the series is convergent or divergent. If it is convergent, find its sum.

1 25

13

n n

n

1

1

)3

(n

n

nn

e

1

1

1n n n

Using partial fractions:

1 A 0 A B

0 1 B

1 B

1

1 1

1n n n

1 1 1 11 1

2 3 3 42

1 1...

1

nS

n n

11

1nSn

lim 1nnS

1

11

A B

n n nn

1 1A n Bn

1 An A Bn

Example

in which nearly every term cancels with a preceding or following term. However, it doesn’t have a set form. Partial fraction decomposition is often used to put in the

above form. Partial sum will be considered since most terms can be

canceled.

Example:

Telescoping Series

...)()()( 433221 bbbbbb

A telescoping series is any series that can be written in the following (or similar) form

12 158

1

n nn

This infinite series converges to 1.

1

11

2

1

21

4

1

4

1

8

1

8

1

161

16

1

32 1

64

1

32

1

64 1

Example

1 2

1

nn

Determine whether the series is convergent or divergent.

In a geometric series, each term is found by multiplying the preceding term by the same number, r.

2 3 1 1

1

n n

n

a ar ar ar ar ar

This converges to if , and diverges if .1

a

r1r 1r

1 1r is the interval of convergence.

Geometric Series

Examples

Determine whether the series is convergent or divergent.

...12

1

4

1

8

1

11

1

5

)6(

nn

n

.3 .03 .003 .0003 .333... 1

3

310

11

10

a

r

3109

10

3

9

1

3

Example: Write 3.545454… as a rational number.

1 4

3

5

2

nn

n

n