Testing convergence of series $\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$ Considering

$$\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$$

depending on $k$, which can be real. I have absolutely no clue how to proceed. Tried to taylor it, but with no result.

Hint. Your series is convergent.
As $n$ tends to $+\infty$, we may write
$$
\begin{align}
u_n &:=\sin \left( \pi \sqrt{n^2+k^2 }\right)\\
&=\sin \left( \pi n \:\sqrt{1+\frac{k^2}{n^2}}\right)\\
&=\sin \left( \pi n \:\left(1+\frac{k^2}{2n^2}+\mathcal{O}\left(\frac{1}{n^4}\right)\right)\right)\\
&=\sin \left( \pi n +\frac{\pi k^2}{2n}+\mathcal{O}\left(\frac{1}{n^3}\right)\right)\\
&=(-1)^n\sin \left(\frac{\pi k^2}{2n}+\mathcal{O}\left(\frac{1}{n^3}\right)\right)\\
&=\frac{\pi k^2}2\frac{(-1)^n}{n}+\mathcal{O}\left(\frac{1}{n^3}\right)
\end{align}
$$
giving the convergence of the initial series $\displaystyle \sum u_n$. 
