Question

In Exercises 37–42, find the series’ radius of convergence.∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ

Accepted Answer

Identify the general term of the series as $$ a_n = \frac{(n!)^2}{2^n (2n)!} x^n . $$Our goal is to find the radius of convergence $$ R  of $$the power series $$ \sum_{n=1}^\infty a_n . $$Use the Ratio Test to find the radius of convergence. Consider the limit $$ L = \lim_{n 	o \infty} \left| \frac{a_{n+1}}{a_n} ight| . $$The radius of convergence $$ R  is $$given by $$ R = \frac{1}{L}  if $$the limit exists. Write out the ratio $$ \left| \frac{a_{n+1}}{a_n} ight| = \frac{\frac{((n+1)!)^2}{2^{n+1} (2(n+1))!} |x|^{n+1}}{\frac{(n!)^2}{2^n (2n)!} |x|^n} = \frac{((n+1)!)^2}{(n!)^2} \cdot \frac{2^n}{2^{n+1}} \cdot \frac{(2n)!}{(2(n+1))!} \cdot |x| . $$Simplify the factorial expressions: $$ \frac{((n+1)!)^2}{(n!)^2} = (n+1)^2 $$, $$ \frac{2^n}{2^{n+1}} = \frac{1}{2} $$, and $$ \frac{(2n)!}{(2n+2)!} = \frac{1}{(2n+1)(2n+2)} . $$Substitute these back into the ratio. Express the limit $$ L = \lim_{n 	o \infty} (n+1)^2 \cdot \frac{1}{2} \cdot \frac{1}{(2n+1)(2n+2)} \cdot |x| . $$Analyze this limit to find $$ L  in $$terms of $$ |x| $$, then solve $$ L \u003c 1  to $$determine the radius of convergence $$ R .$$

In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ

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Key Concepts

Radius of Convergence

Ratio Test for Series Convergence

Factorials and Their Growth