   Chapter 11.6, Problem 10E

Chapter
Section
Textbook Problem

# Use the Ratio Test to determine whether the series is convergent or divergent.10. ∑ n = 0 ∞ ( − 3 ) n ( 2 n + 1 ) !

To determine

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

Theorem:

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

Calculation:

The given series n=1an=n=1(3)n(2n+1)!.

Here, the nth term is, an=(3)n(2n+1)!.

Thus, the (n+1) th term is, an+1=(3)n+1(2(n+1)+1)!.

Obtain the limit of |an+1an|

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