Prove $\sum\frac1{n^p}$ converges only for $p>1$.
For this I need to prove that for $p>1$ and some $n>N\in\mathbb{N}$ $$\frac1{n^p}\le\frac1{p^n}$$
Then I can show that geometric series on RHS is an upper bound. But I am not able to prove this step.
Edit: Above inequality doesn't work as pointed out in the comments. Is there a way to bound this series by some geometric series.