   Chapter 11.7, Problem 9E

Chapter
Section
Textbook Problem

Test the series for convergence or divergence. ∑ n = 0 ∞ ( − 1 ) n π 2 n ( 2 n ) !

To determine

To test:

The series for convergence or divergence.

Explanation

1) Concept:

Series that involves factorials or other products (including a constant raised to the nth power) are often conveniently tested using the ratio test.

Ratio test:

If limnan+1an=L, then n=1an is,

i. Absolutely convergent hence convergent if, L<1

ii. Divergent if L>1 or L=

iii. Test is inconclusive if, L=1

2) Given:

n=0-1nπ2n(2n)!

3) Calculation:

Here,

n=0-1nπ2n(2n)!

The series involves a constant raised to nth term and factorial term therefore, the ratio test should be use.

Consider,

limnan+1an=limn-1

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