<s>[INST] Find all real values of $a$ such that $\sum_{n=0}^{\infty} \sin(\pi\sqrt{a^2+n^2})$ diverges To find all $a$ that makes the series diverge, it is sufficient to make the limit of $\sin(\pi\sqrt{a^2+n^2})$ not exist or exists but not  equal to $0$ as $n\to\infty.$
I suspect only $a=0$ would converge the series, and the limit of $\sin(\pi\sqrt{a^2+n^2})$ would not even exist if $a\neq 0.$ .
 [/INST] HINT:
Observe that we can write
$$\begin{align}
\sin(\pi\sqrt{a^2+n^2})&=\sin\left(n\pi\sqrt{1+\frac{a^2}{n^2}}\right)\\\\
&=\sin\left(n\pi+\frac{a^2\pi}{2n}+O\left(\frac1{n^3}\right)\right)\\\\
&=(-1)^n\sin\left(\frac{a^2\pi}{2n}+O\left(\frac1{n^3}\right)\right)
\end{align}$$
Can you finish now?  Is there any real value for $a$ for which this series diverges?
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