   Chapter 11.R, Problem 44E

Chapter
Section
Textbook Problem

Find the radius of convergence of this series ∑ n − 1 ∞ ( 2 n ) ! ( n ! ) 2 x n

To determine

To find:

The radius of convergence of the given series

Explanation

1) Concept:

The ratio test:

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

2) Given:

n=12n!n!2xn

3) Calculation:

It is given that,

n=12n!n!2xn

Consider,

an=2n!n!2xn

Then an+1  is

an+1=2n+1!n+1!2xn+1

So, by using the ratio test,

limnan+1an=limn2n+1!n+1!2xn+12n!n!2xn

=limn2n+1!xn+1n+1!2·n!22n!xn

=

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