Periodic Function?

Question

What is the period of this function? And how do you come about your answer?

$$\epsilon_n(x)=\sum_{\mu=-\infty}^{+\infty}(x+\mu)^{-n}$$

I understand that $\epsilon_n(x+k)=\epsilon_n(x)$ (sort of)

Answer 1

Hint. By the change of index $\nu=\mu\pm 1$ one gets $$ \epsilon_n(x)=\sum_{\mu=-\infty}^{+\infty}(x+\mu)^{-n}=\sum_{\nu=-\infty}^{+\infty}(x\pm1+\nu)^{-n}=\epsilon_n(x\pm 1), $$ one may prove the period is equal to $1$.