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

You could aswell use the sandwich-lemma:
$$0 < x + \dfrac{x^3}{3} + \dfrac{x^5}{5} + ... < x+x^3+x^5+...<x+x^2+x^3+x^4+...=\sum_n^\infty 1\cdot x^n = \dfrac{1}{1-x}=2$$
Furthermore note that the sum is strictly increasing, hence the sum is convergent.
