   Chapter 11.6, Problem 10E

Chapter
Section
Textbook Problem

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

To determine

Whether the series is convergent or divergent by using Ratio Test.

Explanation

1) Concept:

Use the ratio test.

2) The Ratio Test:

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

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

(iii) If limnan+1an=1, the ratio test is inconclusive, that is, no conclusion can be drawn about the convergence or divergence of an.

3) Given:

n=0(-3)n2n+1!

4) Calculation:

Consider.

n=0an=n=0(-3)n2n+1!

Here an=(-3)n2n+1!  and  an+1=(-3)n+12(n+1)+1!=(-3)n+12n+3!

By the ratio test, consider,

limn

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