Given the number series:
$\begin{aligned}\sum_{n=1}^{+\infty}\end{aligned} (-1)^n \log\left(\frac{2}{\pi}\,\arctan\sqrt{n}\right) $
since both the criterion of Leibniz, is the absolute convergence criteria are inconclusive, I do not know how to proceed.
Some idea?
The series equals:
$$\sum_{n=1}^{\infty} (-1)^{n-1} a_n$$
Where:
$$a_n = \log \left( \frac{\pi}{2\arctan \sqrt n} \right)$$
$a_n$ is nonnegative, decreasing and goes to $0$, so by the alternating series test, the series converges.