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I encountered a problem today to prove that $\cos (n!)$ does not have a limit. I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better.

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closed as off-topic by Johannes Hahn, paul garrett, Andrés E. Caicedo, Wolfgang, quid Dec 24 '15 at 17:40

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It has a limit if the argument of the function is expressed in degrees. – Justin Melvin Nov 2 '10 at 21:01
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The question boils down to whether the sequence $cn!$ tends to a limit mod 1, where $c=1/(2\pi)$. There are transcendental numbers $c$ for which the sequence DOES tend to a limit mod 1, so we have to use something about $\pi$. I'm sorry to see the question closed – SJR Nov 2 '10 at 22:26
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I've started a meta conversation over at tea.mathoverflow.net/discussion/741/does-lim-cosn-exist – David Speyer Nov 3 '10 at 1:08
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IMO it would have made sense for people to participate in the meta thread rather than to have an close/open tug-of-war with no discussion. Oh well. – Ryan Budney Nov 3 '10 at 5:31