   Chapter 11.6, Problem 20E

Chapter
Section
Textbook Problem

# Use the Ratio Test to determine whether the series is convergent or divergent.20. ∑ n = 1 ∞ ( 2 n ) ! ( n ! ) 2

To determine

Whether the series is convergent or divergent.

Explanation

Result used: The Ratio Test

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

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

(ii) If limn|an+1an|=1, the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an.”

Calculation:

The given series n=1an=n=1(2n)!(n!)2.

Here, the nth term is, an=(2n)!(n!)2.

Thus, the (n+1) th term is, an+1=(2(n+1))![(n+1)!]2.

Obtain the limit of |an+1an|.

limn|an+1an|=limn|(2(n+1))![(n+1)!]2(2n)!(n!)2|=limn|(2(n+1))!(n!)2[(<

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