<s>[INST] 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
 [/INST] $$\log(1+x)=x-x^2/2-x^3/3+x^/4-....ad-inf, \log(1-x)=-x-x^2/2-x^3/3-.ad-inf~, if~ |x|<1.$$
Th given series ie quesion is nothing but $\frac{1}{2}\log[(1+x)/(1-x)]$ which is valid (convergent) for $|x|<1$.
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