Using theorems about differentiation or integration of power series calculate infinite sum of
$$ \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)3^n} $$
The answer should equal to $\frac{\pi}{2\sqrt3}$.
I tried using $f(x) = \arctan(x) = \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)}x^{2n+1}$ with $x=\frac{1}{3}$ but that fails, since we have $3^n$ and not $3^{2n+1}$ in the exponent.