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

Question

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 $$ \sum_{n=2}^{\infty}cos(xnln(n)) $$

Answer 1

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