calculus - Finding the sum for the following series

Evaluate

$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)3^n }$$

I can show very easily that this series converges using the alternating series test. By setting

$$b_n = \frac{1}{(2n+1)3^n} \ \ \ \ \ \ \ \Rightarrow b_n \leq b_{n+1}$$

and
$$\lim_{n \to \infty} \frac{1}{(2n+1)3^n} = 0$$

However, what is the sum of the series? I can't find it. I tried to write it out term by term but I don't see any pattern.

Answer

Note that the power series of $\arctan(x)$ is

$$\arctan(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1}$$

Also note that

$$\arctan(\frac{1}{\sqrt{3} }) = \frac{\pi}{6}$$

Therefore,

$$\frac{\pi}{6} = \sum_{n=0}^{\infty} (-1)^n \frac{\big(\frac{1}{\sqrt{3}}\big)^{2n+1}}{2n+1} = \sum_{n=0}^{\infty} \frac{6}{\sqrt{3}} \frac{(-1)^n}{(2n+1)3^n}$$

Thus,

$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)3^n} = \frac{\sqrt{3}\pi }{6}$$