   Chapter 11.6, Problem 9E

Chapter
Section
Textbook Problem

# Use the Ratio Test to determine whether the series is convergent or divergent.9. ∑ n = 0 ∞ ( − 1 ) n − 1 3 n 2 n n 3

To determine

Whether the series is convergent or divergent.

Explanation

Definition used:

“A series an is called absolutely convergent if the series of absolute values |an| is convergent.”

“A series an is called conditionally convergent if it is convergent but not absolutely convergent.”

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, then the Ratio Test is inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an.”

Calculation:

The given series is n=1an=n=1(1)n13n2nn3.

Here, the nth term is, an=(1)n13n2nn3.

Thus, the (n+1) th term is, an+1=(1)n3n+12n+1(n+1)3

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