Monday, November 2, 2015

real analysis - Determine if this series sumlimitsin=1nftyfrac(n!)2(2n)! converges or diverges, and prove your answer?



Determine if this series n=1(n!)2(2n)! converges or diverges, and prove your answer?




I've been able to prove similar problems, but I'm confused now that there's a factorial involved. Can someone help me out here?


Answer



Let an be the n=th term. We use the Ratio Test, and calculate lim. We have
\frac{a_{n+1}}{a_n}=\frac{ \frac{((n+1)!)^2}{(2n+2)!}}{\frac{(n!)^2}{(2n)!}}=\frac{((n+1)!)^2 (2n)!}{(n!)^2(2n+2)!}
Now we start to simplify. Note that \frac{(n+1)!}{n!}=n+1 and \frac{(2n)!}{(2n+2)!}=\frac{1}{(2n+1)(2n+2)}, so our ratio simplifies to
\frac{(n+1)^2}{(2n+1)(2n+2)},
which further simplifies to
\frac{n+1}{2(2n+1)}.
Now find the limit as n\to\infty, perhaps by dividing top and bottom by n. The limit is \frac{1}{4}\lt 1, so we have convergence.


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