# summation of series of powers $n^{nix}$

is the series ..

$$\sum _{n=2}^{\infty}n^{kin}$$

here k is a real number

is convergen or divergent ??, for example perhaps we can copare it to the series $$\sum _{n=2}^{\infty}n^{ix}$$ which is divergent for every fixed 'x'

what criterion could i use to check the convergence or divergence of the series ??

this series is related to the fourier series $$\sum_{n=2}^{\infty}cos(xnln(n))$$

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Does the general term tend to $0$? – 1015 Feb 12 '13 at 22:23
don't know.. i would have for big N the term $\infty ^{i \infty}$ which makes no sense – Jose Garcia Feb 12 '13 at 22:29

Hint: $$|n^{ikn}|=|e^{ikn\ln n}|=1$$ for all $n$. Do you know a criterion that could help now?