How can I prove that for $d>1$ the series converges absolutely, I only know that for $d\leq 0$ the series diverges (In both cases under the assumption that $|z|=1$) What can I say for the other choices for $d$ left.
Is there a way to prove $\sum_{n=0}^{\infty}\frac{1}{n^d}$ converges absolutely for $d>1$ respectively $\sum_{n=0}^{\infty}\frac{1}{n^d}z^n$ for $d>1$ and $|z|=1$ without using the integral test?