Chapter 9.5, Problem 70E

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

# Using a Series In Exercises 69 and 70, use the given series.(a) Does the series meet the conditions of Theorem 9.14? Explain why or why not.(b) Does the series converge? If so, what is the sum? ∑ n = 1 ∞ ( − 1 ) n + 1 a n , a n = { 1 n ,   if   n   is   odd 1 n 3 ,     if   n   is   even

(a)

To determine
The series n=1(1)n+1an , an={1n , if n is odd1n3 , if n is even  satisfies the condition of the

“Alternating series test” i.e. the series n=1(1)n+1an converges if the two conditions satisfy:

limnan=0 and an+1an, for all values of n.

Explanation

Given:

The given series is n=1(1)n+1an , an={1n , if n is odd1n3 , if n is even .

Explanation:

The “Alternating series test” for the series n=1(1)n+1an states that, the series converges if the two conditions holds,

limnan=0 and an+1an, for all n.

For the series to be convergent it must have to satisfy all the condition of “Alternating series test”

(b)

To determine
That the provided series n=1(1)n+1an , an={1n , if n is odd1n3 , if n is even  converges or not. If the the series converges then calculate the sum.

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