   Chapter 11, Problem 25RE

Chapter
Section
Textbook Problem

# Determine whether the series is conditionally convergent, absolutely convergent, or divergent.25. ∑ n = 1 ∞ ( − 1 ) n ( n + 1 ) 3 n 2 2 n + 1

To determine

Whether the series is conditionally convergent, absolutely convergent or divergent.

Explanation

Result used: The Ratio Test

“(i) If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent (and therefore convergent).

(ii) If limn|an+1an|=L>1 or limn|an+1an|= , then the series n=1an is divergent.

(ii) If limn|an+1an|=1 , the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an .”

Calculation:

The nth term of the given series is, an=(1)n1(n+1)3n22n+1 .

Substitute n=n+1 , then an+1=(1)n((n+1)+1)3n+122(n+1)+1 .

Obtain the limit of |an+1an|

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