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.


closed as off-topic by Johannes Hahn, paul garrett, Andrés E. Caicedo, Wolfgang, user9072 Dec 24 '15 at 17:40

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    $\begingroup$ It has a limit if the argument of the function is expressed in degrees. $\endgroup$ – Justin Melvin Nov 2 '10 at 21:01
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    $\begingroup$ The question is now here: math.stackexchange.com/questions/8690/is-there-a-limit-of-cos-n $\endgroup$ – Douglas S. Stones Nov 2 '10 at 21:50
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    $\begingroup$ 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 $\endgroup$ – Sidney Raffer Nov 2 '10 at 22:26
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    $\begingroup$ I've started a meta conversation over at tea.mathoverflow.net/discussion/741/does-lim-cosn-exist $\endgroup$ – David E Speyer Nov 3 '10 at 1:08
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    $\begingroup$ 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. $\endgroup$ – Ryan Budney Nov 3 '10 at 5:31