$S = \frac{\pi}{2} - \ln(2)$

The power series $f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n(2n-1)} x^{2n-1}$ has a radius of convergence $R = 1$.

$x f'(x) + f(x) = 2 \arctan(x)$ for $|x| < 1$

The series converges absolutely at both endpoints of its interval of convergence.