is this sequence : $\Big(1+\Big(\frac{(-1)^n}{n}\Big)\Big)^{n(-1)^n}$ divergent ? I notice that when $n$ is even it converges to $e$ and when $n$ is odd is converges to $e$ as well but I've seen somewhere that this sequence is divergent. how is that ?
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1Where did you see that it does not converge? If both even subsequences and odd subsequences converge to the same value, then the original sequence does what? – Shashi Dec 08 '17 at 15:09
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has to be convergent – lion Dec 08 '17 at 15:12
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Odd subsequence has the form $$\left[\left(1-\frac{1}{n}\right)^n\right]^{-1}\to \left(e^{-1}\right)^{-1}=e.$$ – szw1710 Dec 08 '17 at 15:21
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Please consider to modify the title, it's misleading with respect to the asked question. – user Dec 08 '17 at 15:41
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If you could show $|a_n-e| \lt \frac{2}{n}$ for integer $n \gt 1$, would that persuade you the sequence converged to $e$? Could you show this or something similar for the two subsequences? – Henry Dec 08 '17 at 16:24
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Please, if you are ok, you can accept the answer and set it as solved. Thanks! – user Jan 24 '18 at 21:53
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Try to show that if $a_{2n}\to a$ and $a_{2n-1}\to a$ then $a_n\to a$. This is an easy observation.
szw1710
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