I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). 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. Thanks
Is there a limit of cos (n!)?
28
$\begingroup$
analysis
trigonometry
-
0Do you mean the sequence (X_n) with X_n = cos(n!), or are you referring to a function f(n) = cos(n!) ? Also, is this homework? – 2010-11-02
-
0No, this is not a homework, just my curiosity (from a simple exercise about limits where we had cos(n!), but the limit of it didn't matter). – 2010-11-02
-
3@Cam: those are equivalent. – 2010-11-02
-
0In light of David's answer, can you mention where you found this problem, and why you think it is a true statement? – 2010-11-02
-
0Why cos(n!) would have a limit? It is oscillating back and forth from -1 to 1. – 2010-11-02
-
2@Robert: as David's answer indicates, although it is intuitive that cos(n!) should be oscillating, actually proving it seems to require that we know much more about certain properties of pi than we actually know. – 2010-11-03
-
0@Qiaochu: That's very interesting. Do we have enough information to prove that the limit doesn't exist? Thank you for the clarification. – 2010-11-03