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

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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. –  rajkamalmath Jan 18 '13 at 9:22
    
Hint: What is the closest integer to $en!$? –  Sanchez 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! –  rajkamalmath 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
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marked as duplicate by amWhy, Claude Leibovici, Michael Hoppe, Asaf Karagila, egreg Mar 22 at 15:28

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1 Answer

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 $2\pi$. –  rajkamalmath Jan 18 '13 at 10:05
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