LESSON 70 – Alternating Series and Absolute Convergence & Conditional ConvergenceHL Math –Santowski

OBJECTIVES

(a) Introduce and work with the convergence/divergence of alternating series

(b) Introduce and work with absolute convergence of series

(c) Deciding on which method to use ……

series diverges

Converges to a/(1-r)if |r|<1. Diverges if

|r|>1

Converges if p>1Diverges if p<1

non-negativeterms and/orabsoluteconvergence

Does Σ |an| converge?Apply Integral Test, RatioTest or nth-root Test

Original SeriesConverges

AlternatingSeries Test

Is Σan = u1-u2+u3-…an alternating series

Is there an integer Nsuch that uN>uN-1…?

Converges if un 0Diverges if un 0

nth-Term Test Is lim an=0 no

no

yes or maybe

no

no or maybe

GeometricSeries Test Is Σan = a+ar+ar2+ … ?

yes

yes

yes

yes

yes

p-Series Test Is series form

1

1

npn

.PROCEDURE FOR DETERMINING CONVERGENCE

ALTERNATING SERIESA series in which terms alternate in sign

1

( 1)n nn

a

1

1

( 1)n nn

a

or

1

1

1 1 1 1 1( 1) ...

2 4 8 162n

nn

1

1 1 1 1( 1) 1 ...

2 3 4n

n n

TESTING CONVERGENCE - ALTERNATING SERIES TEST

...)1( 3211

1

uuuunn

n

1. each un is positive;2. un > un+1 for all n > N for some integer N (decreasing);3. lim n→∞ un@ 0

Theorem The Alternating Series Test

The series

converges if all three of the following conditions are satisfied:

Alternating Series

example: 1

1

1 1 1 1 1 1 11

1 2 3 4 5 6n

n n

TESTING CONVERGENCE - ALTERNATING SERIES TEST

Alternating Series

example: 1

1

1 1 1 1 1 1 11

1 2 3 4 5 6n

n n

This series converges (by the Alternating Series Test.)

If the absolute values of the terms approach

zero, then an alternating series will always

converge!

Alternating Series Test

This series is convergent, but not absolutely convergent.

Therefore we say that it is conditionally convergent.

TESTING CONVERGENCE - ALTERNATING SERIES TEST

EXAMPLES

Investigate the convergence of the following series:

Show that the series converges

(a) ( 1)n11n2

n1

(b) ( 1)n11n3

n1

(c) ( 1)n1n1

nn1

(-1)n+1 ln(n)nn2

ABSOLUTE AND CONDITIONAL CONVERGENCE A series is absolutely convergent if the

corresponding series of absolute values

converges.

A series that converges but does not converge absolutely, converges conditionally.

Every absolutely convergent series converges. (Converse is false!!!)

nn N

a

nn N

a

IS THE GIVEN SERIES CONVERGENT OR DIVERGENT? IF IT IS CONVERGENT, ITS IT ABSOLUTELY CONVERGENT OR CONDITIONALLY CONVERGENT?

( 1) / 2

1

( 1)

3

n n

nn

1

( 1)

ln( 1)

n

n n

1

( 1)n

n n

1

1

( 1) ( 1)n

n

n

n

A) IS THE GIVEN SERIES CONVERGENT OR DIVERGENT? IF IT IS CONVERGENT, ITS IT ABSOLUTELY CONVERGENT OR CONDITIONALLY CONVERGENT?

( 1) / 2

1

( 1) 1 1 1 1...

3 9 27 813

n n

nn

This is not an alternating series, but since

( 1) / 2

1 1

( 1) 1

3 3

n n

n nn n

Is a convergent geometric series, then the givenSeries is absolutely convergent.

B) IS THE GIVEN SERIES CONVERGENT OR DIVERGENT? IF IT IS CONVERGENT, ITS IT ABSOLUTELY CONVERGENT OR CONDITIONALLY CONVERGENT?

1

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

ln( 1) ln 2 ln3 ln 4

n

n n

Diverges with direct comparison with the harmonicSeries. The given series is conditionally convergent.

1

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

ln( 1) ln 2 ln3 ln 4

n

n n

Converges by the Alternating series test.

C) IS THE GIVEN SERIES CONVERGENT OR DIVERGENT? IF IT IS CONVERGENT, ITS IT ABSOLUTELY CONVERGENT OR CONDITIONALLY CONVERGENT?

1

1

( 1) ( 1) 2 3 4 5

1 2 3 4

n

n

n

n

By the nth term test for divergence, the seriesDiverges.

D) IS THE GIVEN SERIES CONVERGENT OR DIVERGENT? IF IT IS CONVERGENT, ITS IT ABSOLUTELY CONVERGENT OR CONDITIONALLY CONVERGENT?

1

( 1) 1 1 1 1

1 2 3 4

n

n n

Converges by the alternating series test.

1

( 1) 1 1 1 1

1 2 3 4

n

n n

Diverges since it is a p-series with p <1. TheGiven series is conditionally convergent.

FURTHER EXAMPLES