Is the infinite series $\sum_n n^{i\theta}$ convergent for any values of $ \theta$? > Is the infinite series $\sum_n n^{i\theta}$ convergent for any values of $\theta$? I have seen tricks for converting $\sum \sin(n) $ into a telescoping series, but am stumped because $ n^{i\theta} = e^{i \theta \ln n} = \cos (\theta \ln n) + i \sin (\theta \ln n) $ doesn’t seem like could succumb to similar trick. Any ideas? Thanks!

For it to converge, both real part and imaginary part should converge to some values, respectively. It seems like for all value of $theta$, neither is converging.

For a $\sum \cos(\theta \ln n)$ to converge, a necessary condition is $\cos(\theta \ln n)$ has $0$ as its limit which it apparently doesn't even when $\theta=0$.