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This question already has an answer here:

Please give me some hint to proceed. I'm clueless:

Show that, $\lim\limits_{n\to \infty}n\sin(2\pi en!)=2\pi$

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marked as duplicate by Martin Sleziak, Marc van Leeuwen, Davide Giraudo, drhab, Sami Ben Romdhane Jan 23 '15 at 13:15

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
up vote 5 down vote accepted

Note that $e=\sum_{k=0}^\infty\frac1{k!}$, hence $en!=\sum_{k=0}^n\frac{n!}{k!}+\frac1{n+1}+\ldots$ is quite close to an integer.

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