Does the Series Converge?1k

k

a

10 Tests for Convergencenth Term Divergence Test

Geometric Series

Telescoping Series

Integral Test

p-Series Test

Direct Comparison Test

Limit Comparison Test

Alternating Series Test/Absolute Convergence Test

Ratio Test

Root Test

Each test has it limitations (i.e. conditions where the test fails).

The test tells you nothing!

11.5A Alternating Series – terms alternate in signs

11 2 3 4

1

( 1) nn

n

a a a a a

1 2 3 41

( 1) nn

n

a a a a a

OR

NOTE: All an’s are assumed to be positive.

OTHER FORMS OF ALTERNATING SERIES

Instead of using to create an alternating series, can be used.

Be careful… a series with both positive and negative terms is not alternating unless every other term alternates between positive and negative, with the absolute value of all terms being generated with the same rule for .

cos( ) n1( 1) or ( 1)n n

na

Alternating Series - Examples

1

1

( 1) 1 1 11

2 3 4

n

n n

The alternating harmonic series (will prove to be

convergent).

1

( 1) 1 1 1

2 2 4 8

n

nn

An alternating geometric series (convergent because

r = –1/2).

1

(cos( )) 1 2 3 4 n

n n

A divergent alternating series (nth-term test).

The Alternating Series Test

11 2 3 4

1

( 1) nn

n

a a a a a

The series …

Converges if …

12. a , for some n na n N N

3. lim 0nna

1. 0 na n

The Alternating Series Test

1

1

( 1)n nn

a

Converges if …

“Proof”:

1. a 0 n n

12. a n na 3. lim 0n

na

Alternating Series - Examples

1

1

( 1)

2 3

n

n

n

n

1

1

( 1)n n

n

ne

1

1

( 1) 1 1 11

2 3 4

n

n n

11.5B Approximating Alternating Series

If an alternating series satisfies the conditions of the alternating series test, and SN , the partial sum of the first N terms, is used to approximate the sum, S; then …

1

1 1

, orN N N

N N N N

S S R a

S a S S a

The error, RN, is less than the first term omitted.

Approximating Alternating Series

Example:

1. Determine the sum of the first 4 terms.

1

1

( 1)

!

n

n n

14

1

( 1) 1 1 1 10.625

! 1 2 6 24

n

n n

Approximating Alternating Series

Example: 1

1

( 1)

!

n

n n

5 1( 1) 1.008333

5! 120

2. Estimate the error if 4 terms are used to approximate the sum.

3. Therefore the sum, S, lies between:

0.625 0.008333 0.625 0.008333, or

0.616667 S 0.633333

S

Approximating Alternating Series

Example: 1

1

( 1)

!

n

n n

2. How many terms are needed to make sure the

error is less than 0.01?

( 1) 1( 1)0.01

( 1)!

n

n

1

.0416664!1

.0083335!

10.01

( 1)!n

Therefore, four terms are needed!

Example 1 of the Alternating Series Test

1 1 1 1 1( 1) 1

2 3 4n

n

Decreasing?

2

1 10 when 0

dx

dx x x

Limit?1

lim 0n n

Therefore, convergent.

The Alternating Harmonic Series

Example 2 of the Alternating Series Test

1 ln( 1)n

n

n

Decreasing?

2 2

ln (1/ ) ln 1 ln0 when 3

d x x x x xx

dx x x x

Limit?ln 1/

lim lim 01n n

n n

n

Therefore, convergent.

Absolute Convergence

1. na converges absolutely …

na

converges.if

2. na converges conditionally …

if

| |na

| |naconverges but...

diverges.

Absolute Convergence: Example 1

1 1 1 1 1 1 11

2 3 4 5 6 7 8na

1na n

Divergent harmonic series, therefore the alternating series is conditionally convergent but not absolutely convergent.

is a convergent alternating series.

Absolute Convergence: Example 2

2 2 2 2

1 1 1 1 11

4 9 16 25 361 1 1 1

, lim 0 and .( 1)

n

nn

a

an n n n

2

1na n

Convergent p-series, therefore the alternating series is absolutely convergent.

is a convergent alternating series.na

Absolute Convergence: Example 3

1 1 1 1 1 1 11

2 4 8 16 32 64 128na

1

1

2n na Convergent geometric series,

therefore the first series converges absolutely.

Therefore, the original series converges.

Absolute Convergence Test: Ex. 1

1 1 1 1 1 1 11

2 4 8 16 32 64 128na

1

1

2n na Convergent geometric series,

therefore the first series converges absolutely.

If a series converges absolutely, it is a convergent series.

Note that the first series is NOT an alternating series.