Power series summation Trying to find the sum of the following infinite series:
$$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$
Any ideas on how to find this sum?

Note that for suitable $t$,
$$\frac{1}{1+t^2}=1-t^2+t^4-t^6+\cdots.$$
Integrate term by term from $0$ to $x$. We get
$$\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots.$$
Divide by $x$. We get
$$\frac{\arctan x}{x}=1-\frac{x^2}{3}+\frac{x^4}{5}-\frac{x^6}{7}+\cdots.$$
Finally, let $x=\frac{1}{\sqrt{3}}$.
