Convergence of series $x+\frac{x^3}{3}+\frac{x^5}{5}+\cdots$ Convergence of series $$S=x+\frac{x^3}{3}+\frac{x^5}{5}+\cdots$$
When $x=\frac{1}{2}$
I used ratio test as:
$$a_{n}=\frac{\left(\frac{1}{2}\right)^{2n-1}}{2n-1}$$
Then
$$a_{n+1}=\frac{\left(\frac{1}{2}\right)^{2n+1}}{2n+1}$$
Then we get:
$$\frac{a_{n+1}}{a_n}=\frac{2n-1}{4(2n+1)}$$
Hence we get:
$$\lim_{n \to \infty}\frac{a_{n+1}}{a_n}=0.25 \lt 1$$
Hence $\sum a_n$ Converges.
But how to find its value, i guess $\arctan x$ is not useful here

Take derivative of $S$ to get $$S'=1+x^2+x^4+...=\frac {1}{1-x^2}$$
Integrating and you get $$S=(1/2)\ln\left|\frac {1+x}{1-x}\right|$$ 
At $x=1/2$ the result is $\frac {\ln 3}{2}$
