   Chapter 11, Problem 44RE

Chapter
Section
Textbook Problem

# Find the radius of convergence of the series ∑ n = 1 ∞ ( 2 n ) ! ( n ! ) 2 x n

To determine

To find: The radius of convergence and interval of convergence of the series.

Explanation

Given:

The series is n=1(2n)!xn(n!)2 .

Result used:

(1)Ratio test:

If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent.

Calculation:

Let an=(2n)!xn(n!)2 .

Then an+1=(2(n+1))!xn+1((n+1)!)2 .

Obtain |an+1an| to apply the Ratio test.

|an+1an|=|(2(n+1))!xn+1((n+1)!)2(2n)!xn(n!)2|

Take limn on both sides:

limn|an+1an|=limn|(2(n+1))!xn+1((n+1)!)2(2n)!xn(n!)2|=limn|(2(n+1))!xn+1((n+1)!)2(n!)2(2n)!xn|=limn|(2n+2)!x(<

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