series convergence test with parameter As part of a bigger proof I reached the series $\sum {1 \over n^\alpha}$. $\alpha \in \mathbb{R}$   
Obviously, the convergence depends on the value of $\alpha$. 
I already know the harmonic series diverges while $\sum {1\over n^2}$ converges.  
How to solve the general case?  

Some notes:


*

*Because $\sum \tfrac 1n$ diverges you can prove the divergence for $\alpha \le 1$ with the limit comparison test

*Because $\sum \tfrac 1{n^2}$ converges you can prove the convergence for $\alpha \ge 2$ with the limit comparison test

*Concerning an article I have read, you can use the Cauchy condensation test to prove the divergence and convergence of the series.

*The function $(1,\infty)\rightarrow \mathbb R: \alpha \mapsto \sum \tfrac1{n^\alpha}$ is the famous Riemann zeta function.

