<s>[INST] Convergence of the series $\sum_{n = - \infty}^{\infty} (\sqrt{(a+n)^2+b^2} - |n| )$ Given $a, b \in \mathbb{R}$ with $b > 0$, is the series
\begin{equation}
\sum_{n = - \infty}^{\infty} (\sqrt{(a+n)^2+b^2} - |n| )
\end{equation}
convergent or divergent? 
If we drop out the $n=0$ term and fold the remaining sum, the question can be equivalently asked for the series
\begin{equation}
\sum_{n = 1}^{\infty} (\sqrt{(a+n)^2+b^2} + \sqrt{(a-n)^2+b^2} - 2n ).
\end{equation}
 [/INST] $$
\begin{align}
\lim_{n \to \infty }\sqrt{a^2+2an+n^2+b^2}-n=\lim_{n \to \infty }\frac{\left ( a^2+2an+b^2 \right )}{\sqrt{a^2+2an+n^2+b^2}+n}=a
\end{align}
$$
evidently divergent for 
$$
\begin{align}
a\neq 0
\end{align}
$$
</s>