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Given the series: $\sum_{n=0}^{\infty}(-1)^{n}(\sqrt[n]{n} - 1)^{n}$. Does the series converge?

Attempt to solution (might be incorrect):

$(\sqrt[n]{n} - 1)^{n}> (1+\frac{1}{n})^{n}$

$(1+\frac{1}{n})^{n} \to e \Rightarrow (\sqrt[n]{n} - 1)^{n}$ lower-bounded by $e$. Based on Leibniz Criterion the sequence $\{A_n\}$ (in our case, $(\sqrt[n]{n} - 1)^{n}$) is monotone decreasing, but its limit is not $0$ at infinite $\Rightarrow$ series diverge.

Is it enough to say that since the sequence is lower-bounded, the limit of it at infinite is not $0$, or should I actually calculate the limit of the sequence?

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    The limit of $(-1)^n (n^{1/n} - 1)^n$ goes to $0$.2010-11-27
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    To see that your inequality is false, note that it is equivalent to $\root n\of n -1>1+(1/n)$, or $n>(2+(1/n))^n$. But $(2+(1/n))^n>2^n>n$.2010-11-27

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