February 15th, 2019, 01:32 PM  #1 
Banned Camp Joined: Feb 2019 From: Casablanca Posts: 23 Thanks: 2  weird series
Hello everyone Here is a $Z_n$ Suite $\displaystyle Z_3 = 1/2$ $\displaystyle Z_n = Z_ {n + 1} / \cos ({\pi} / {n})$ It's a bizarre suite decreasing to 0 without ever going. I will define the n with which I work in my suite so as not to fall into absurdities like 1/0 which gives false calculations. find that $\displaystyle n = \pi / \arccos (Z_{n + 1} / Z_n) $ Note that $Z_n \ne 0$ and $(Z_{n + 1} / Z_n) \ne 1$, so the $Z_n$ limit is nonzero. And since I have a decreasing sequence, $\displaystyle Z_{n + 1} / Z_n < 1$, so the series $Z_n$ is convergent according to the d'Alembert criterion. Is there an error in this reasoning? $Z_n$ is a positive term and is decreasing and minus 0 tends to 0 minus bound 0, which is impossible because $Z_n \ne 0$, no? Last edited by skipjack; February 15th, 2019 at 01:48 PM. 

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