Alternating Series

Lesson 9.5

Alternating Series

Two versions

• When odd-indexed terms are negative

• When even-indexed terms are negative

21

1 3( 1) ...kk

k

a aa a

11 3

12( 1) ...k

kk

aaa a

Alternating Series Test

• Recall does not guarantee convergence of the series

In case of alternating series …

• Must converge if•

• { ak } is a decreasing sequence(that is ak + 1 ≤ ak for all k )

lim 0kka

lim 0kka

Alternating Series Test

• Text suggests starting out by calculating

• If limit ≠ 0, you know it diverges• If the limit = 0

• Proceed to verify { ak } is a decreasing sequence

• Try it …

lim kka

1

21

( 1)

1

k

k

k

k

1

( 1)k

k k

Using l'Hopital's Rule

• In checking for l'Hopital's rule may be useful

• Consider

• Find

lim kka

1

1

( 1) lnk

k

k

k

ln 1/lim lim 0

1x x

x x

x

lim ?kka

Absolute Convergence

• Consider a series where the general terms vary in sign • The alternation of the signs may or may not

be any regular pattern

• If converges … so does

• This is called absolute convergence

ka

ka ka

Absolutely!

• Show that this alternating series converges absolutely

• Hint: recall rules about p-series

1

3/ 21

( 1)k

k k

Conditional Convergence

• It is still possible that even thoughdiverges …

• can still converge

• This is called conditional convergence

• Example – consider vs.

ka

ka

1

( 1)k

k k

1

1

k k

Generalized Ratio Test

• Given

• ak ≠ 0 for k ≥ 0 and

• where L is real or

• Then we know• If L < 1, then converges absolutely• If L > 1 or L infinite, the series diverges• If L = 1, the test is inconclusive

ka1lim k

kk

aL

a

ka

Apply General Ratio

• Given the following alternating series• Use generalized ratio test

2 11

1

2( 1)

!

kk

k k

Assignment

• Lesson 9.5

• Page 636

• Exercises 1 – 29 EOO