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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14$\begingroup$ It has a limit if the argument of the function is expressed in degrees. $\endgroup$– Justin MelvinCommented Nov 2, 2010 at 21:01
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4$\begingroup$ It may not be research level, but then again, it's a little bit of a tough call as to what qualifies as a research question. I agree with David that this problem might not be trivial, but closely related problems (such as showing that $\cos(n)$ has no limit) are well-trodden ground. $\endgroup$– Todd TrimbleCommented Nov 2, 2010 at 22:17
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17$\begingroup$ 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 RafferCommented Nov 2, 2010 at 22:26
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6$\begingroup$ I've started a meta conversation over at tea.mathoverflow.net/discussion/741/does-lim-cosn-exist $\endgroup$– David E SpeyerCommented Nov 3, 2010 at 1:08
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4$\begingroup$ 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 BudneyCommented Nov 3, 2010 at 5:31
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