1

Lecture 28 – Alternating Series Test

Goal: Does a series (of terms that alternate between positive and negative) converge or diverge?

13

21

11

n

n

n

n

1

11

k

k

k

1

2

11n

kn

k

e

...126

25

65

16

28

9

9

4

2

1

...5

1

4

1

3

1

2

11

...251694

5432

eeee

e

1

11n

nn a ...

8

1

3

1

4

1

2

1

2

11

2

Alternating Series Test

A series of the form with

is called alternating and converges if:

1

11n

nn a 0na

Example 1 – Converge/Diverge?

3

1

lim3

2

n

n

n

1)(

3

2

x

xxf

Thus,

Nna

aa

n

nn

allfor 0lim

.2

.1

n

1

13

21

11

n

n

n

n

is f Since

Example 2 – Converge/Diverge?

4

n

n

1lim

Nna

aa

n

nn

allfor 0lim

.2

.1

n

1

1

11

k

k

k

Example 3 – Converge/Diverge?

5

2

limn

e

n

n

Nna

aa

n

nn

allfor 0lim

.2

.1

n

1

1

2

11k

kk

k

e

Example 4 – Converge/Diverge?

6

n

na lim

Nna

aa

n

nn

allfor 0lim

.2

.1

n

1

1

11n

nn a ...

8

1

3

1

4

1

2

1

2

11

even is if

odd is if

2

11

2

2/n

n na

n

n

Using Sn (nth partial sum) to approximate S

7

n R

...)1()1(... 121

4321

nn

nn aaaaaaS

...)1( 12

n

nn aS

The remainder term when using Sn:

Example 5

8

How many terms are needed to approximate the series and yield a remainder less than 0.0001?

...36

1

25

1

16

1

9

1

4

11

12

11

n

n

n

Absolutely Converge, Conditionally Converge, or Diverge

9

.absolutely converges then converges, If nn bb

1

1

1

1n

nn

nn ab

diverges. then diverges, If nn bb

lly.conditiona convergesthen

diverges, and converges If

n

nn

b

bb

Example 6 – AC, CC, D?

10

Look at absolute value:

12

11

n

n

n

12

1

n n

2

1lim

n

n

Example 7 – AC, CC, D?

11

Look at absolute value:

1

11

n

n

n

1

1

n n

n

n

1lim

Example 8 – AC, CC, D?

12

Look at absolute value:

1

11 tan1n

n n

1

1tann

n

n

n

1tanlim