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.
No comments:
Post a Comment