Consider the series
\[ \sum ^{\infty }_{n=1} \frac {(n!)^2}{(2n)!} \]
We want to determine whether the series converges or
diverges.
Which of the following methods would be the best to determine convergence of the series:
integral test root test geometric series telescoping series ratio test
We will use the Ratio Test to test for convergence of the series. We need to calculate the ratio
\[ \frac {a_{n+1}}{a_n} \]
where \(a_{n+1}=\answer { \frac {((n+1)!)^2 }{(2(n+1))!}}\) and \(a_n=\answer { \frac {(n!)^2}{(2n)!}}\).
Consider the ratio \(\frac {a_{n+1}}{a_n}\). In order to simplify this expression we simplify the following
pieces:
\( \frac {((n+1)!)^2}{(n!)^2}=\answer {(n+1)^2}\) and \(\frac { (2n)!}{(2(n+1))!}=\answer {\frac {1}{(2n+1)(2n+2)}}\)
Simplifying we obtain \( \frac {a_{n+1}}{a_n} =\answer { \frac {n+1}{2(2n+1)}}\)
Now we need to calculate \(\lim _{n\to \infty } \frac {a_{n+1}}{a_n} =\answer { \frac {1}{4} }\)