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Mathematics 22 Online
OpenStudy (watchmath):

Compute \[\lim_{n\to \infty}\frac{n!}{2^{n^2}}\]

OpenStudy (anonymous):

I believe that \(2^{n^2} \) grows faster than \(n!\).

OpenStudy (watchmath):

How if you "show" that? :)

OpenStudy (anonymous):

Hmm I would think of using the squeeze theorem somehow to show that.

OpenStudy (anonymous):

let \[t_{n}=\frac{n!}{2^{n^2}}\]\[t_{n+1}=\frac{(n+1)!}{2^{(n+1)^2}}=\frac{(n+1)}{2^{2n+1}}t_{n}\]\[t_{n+1}=\frac{2(n+1)}{2^{2(n+1)}}t_{n}\]now obviously 2^2(n+1) grows faster than 2(n+1) so the series converges...

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