Do the convergent sum
$$\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n^2+a^2}}$$
posses a closed form? ($a \in \mathbb{R}$)
Special case is known, for $a=0$ one recalls well known alternating harmonic series :
$$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}=-\ln 2$$