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Calculus / Calculus: Early Transcendentals 1 / Chapter 8.5 / Problem 17E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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The Ratio Test Use the Ratio Test to determine

Chapter 12, Problem 17E

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QUESTION:

9-18. The Ratio Test Use the Ratio Test to determine whether the following series converge.

\(\sum_{k=1}^{\infty} \frac{(k !)^{2}}{(2 k) !}\)

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QUESTION:

9-18. The Ratio Test Use the Ratio Test to determine whether the following series converge.

\(\sum_{k=1}^{\infty} \frac{(k !)^{2}}{(2 k) !}\)

ANSWER:

Problem 17E

The Ratio Test
Use the Ratio Test to determine whether the following series converge.
$\sum\limits_{k=1}^{\infty{}}\frac{{\left({k!}\right)}^{2}}{\left({2k}\right)!}$
Solution:
Step 1
We compute the limit L,
$L=\lim\limits_{{k} \rightarrow {\infty{}}}\left|{\frac{{a}_{k+1}}{{a}_{k}}}\right|$
${a}_{k+1}=\frac{{\left({(k+1)!}\right)}^{2}}{\left({2(k+1)}\right)!}$ and ${a}_{k}=\frac{{(k!)}^{2}}{\left({2k}\right)!}$

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