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

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marked as duplicate by amWhy, Claude Leibovici, Michael Hoppe, Asaf Karagila, egreg Mar 22 '14 at 15:28

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Sentences "evaluate $x$" won't help you receive advises from people. Especially if you use exclamation marks. What have you tried? – Ilya Jan 18 '13 at 9:16
@Ilya: I have tried but cannot find anything. – asimath Jan 18 '13 at 9:22
Hint: What is the closest integer to $en!$? – user27126 Jan 18 '13 at 9:28
I can tell the answer to the question is $2\pi$. I dont know how to do it. Help! – asimath Jan 18 '13 at 9:37
@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 9:37
up vote 4 down vote accepted

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

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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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