Evaluate $\lim_{n\to\infty}n\sin(2\pi \mathrm en!)$

Question

Evaluate $$\lim_{n\to\infty}n\sin(2\pi \mathrm en!)$$

Sentences "evaluate $x$" won't help you receive advises from people. Especially if you use exclamation marks. What have you tried? — SBF, Jan 18 '13 at 09:16
I can tell the answer to the question is $2\pi$. I dont know how to do it. Help! — asimath, Jan 18 '13 at 09:37
@Ilya: not sure if you were being facetious, but the exclamation mark is a factorial, not an emotional outburst. — Ron Gordon, Jan 18 '13 at 09:37
@rlgordonma: neither I'm sure, but in a way OP is stated, it's more likely to be an exclamation mark (which fits better the style) than the factorial. — SBF, Jan 18 '13 at 09:39

score 4 · Accepted Answer · answered Jan 18 '13 at 09:55

4

Hint: If you write $${\rm e}=\sum_{k=0}^\infty {1\over k!}$$then $${\rm e}n!=({\rm integer})+{1\over n+1}+({\rm small\ number})$$

answered Jan 18 '13 at 09:55

Per Erik Manne

I think if we solve using this result we will get answer $2\pi$. – asimath Jan 18 '13 at 10:05

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