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What happens when checking for power of convergence, the limit does not exist? I have this problem $(1+\frac{(-1)^n}n)^{n^2}\cdot\frac{(2x+1)^n}{n}$

If I try Cauchy's root test, I get the following

$\lim: \lim_{n\to 0} (1+\frac{(-1)^n}n)^{n}\frac{|2x+1|}{^n\sqrt{n}}$

The problem is that the limit does not exist, as appraoching $n$ with the sequence $2k+1$ or $2k$ yields different results.

However according to wolfram-alpha, it exists and [equals one](http://www.wolframalpha.com/input/?i=lim+as+n+approaches+infinity+(1%2B(-1)%5En%2Fn)%5En)!

So, the question is, does the limit exist and it equals one? or does it not exist at all? and moreover, what happens when the limit does not exist in Cauchy's test? if it does not, how do I solve the question?

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    I also tried applying d'alembert's ratio test. I hit the same problem. According to everything I learnt in mathamatics the limit of the first product should not exist as it is approaching two different values in e and 1/e, however it seems to be the product of the two values, am I missing something here or what exactly? also, I know n^(1/n) approaches 1 and I proved it for this question, however I can't get which of the limit to take, e? 1/e? or 1? – 2017-02-18

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Try $\limsup$ version of Cauchy-Hadamard theorem.

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    I am not sure I understood that, is it the supremum of all subsequences? in this case I have two, e and e^-1 so I choose e? – 2017-02-18
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    Of course, it's that. $\limsup$ is a supremum of limits of all convergent subsequences. Are you sure there are only two convergent subsequences? But it is easy to prove that limes superior really equals to $e$. – 2017-02-18