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Extrazlove February 15th, 2019 01:32 PM

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 non-zero.
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?


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