Let $x \in \mathbb{R} \setminus \mathbb{Z}$ and
$$f(x)=\sum_{k=1}^\infty\frac{1}{(x-k)^2}+\sum_{k=1}^\infty\frac{1}{(x+k)^2}+\frac{1}{x^2}$$
I need to show that $f(x)$ is well-defined and continuous on $\mathbb{R} \setminus \mathbb{Z}$.
The hint our professor gave us was to look at a $x \in [-K, K]$, $K \in \mathbb{N}$, and write $f(x)$ as
$$f(x)=\sum_{k=K+1}^\infty\frac{1}{(x-k)^2}+\sum_{k=K+1}^\infty\frac{1}{(x+k)^2}+\sum_{k=-K}^K\frac{1}{(x-k)^2}$$ and then show that the Weierstrass-M-test https://en.wikipedia.org/wiki/Weierstrass_M-test can be apllied to both sequences. Does anyone have any idea how to do this using the hint described above (so without Herglotz trick for example) and how to continue after showing that the M-test can be apllied?
Take $x\in\mathbb{R}\setminus\mathbb Z$. Fix a natural number $K>|x|$. Then $x\in[-K,K]$. Then$$\sum_{k=K+1}^\infty\left|\frac1{(x-k)^2}\right|\leqslant\sum_{k=1}^\infty\frac1{(k-K)^2}<+\infty.$$The same thing holds for $\sum_{k=K+1}^\infty\frac1{(x+k)^2}$. So, your hint exrpesses $f(x)$ as the sum of $3$ functions, each of which is continuous at $x$.