   Chapter 11.6, Problem 21E

Chapter
Section
Textbook Problem

Use the Ratio Test to determine whether the series is convergent or divergent. 1 − 2 ! 1 ⋅ 3 + 3 ! 1 ⋅ 3 ⋅ 5 − 4 ! 1 ⋅ 3 ⋅ 5 ⋅ 7 + ... + ( − 1 ) n − 1 n ! 1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ ⋅ ⋅ ( 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, then the ratio test is inconclusive, that is, no conclusion can be drawn about the convergence or divergence of an.

3) Given:

1-2!1·3+3!1·3·5-4!1·3·5·7+ +-1n-1n!1·3·5· · · ·2n-1+ · · ·

4) Calculation:

Consider.n=1an=n=1-1n-1 n!1·3·5· · · ·2n-1

Here,an=-1n-1 n!1·3·5· · · ·2n-1  and  an+1=-1n (n+1)!1·3·5· · · ·2n-12n+1

By the ratio test, consider,

limnan+1an= limnx

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