Show that $\lim_{n\rightarrow \infty}\sum_{i=0}^n \bigl(\frac{1}{i\pi+\frac{\pi}{2}}\bigr)^r$ converges only when $r > 1$

Question

Show that $\lim_{n\rightarrow \infty}\sum_{i=0}^n \left(\frac{1}{i\pi+\frac{\pi}{2}}\right)^r$ converges only when $r > 1$.

I can prove that the expression doesn't converge when $r \leq 1$ but I don't know how to show that it converges when $r >1$.

Could you give any hints?

Answer 1

Using $k$ instead of $i$ for the index of summation (because $i$ can sometimes be confused with $\sqrt{-1}$), we have

$$\sum_{k=0}^n\left(1\over k\pi+\pi/2 \right)^r=\left(2\over\pi\right)^r+\sum_{k=1}^n\left(1\over k\pi+\pi/2 \right)^r\lt\left(2\over\pi\right)^r+{1\over\pi^r}\sum_{k=1}^\infty{1\over k^r}$$

and the final infinite series converges when $r\gt1$.