Sum of $\sum\limits_{i=1}^\infty (-1)^i\frac{x^{2i+1}}{2i+1}$ Can someone help me with this series? It was on my exam and I don't know how to do it.
For $|x| < 1$ determine the sum of
$$\sum\limits_{i=1}^\infty  (-1)^i\frac{x^{2i+1}}{2i+1}$$

The derivative of the given sum is the geometric sum
$$\sum_{n=1}^\infty  (-x^2)^n=-\frac{x^2}{1+x^2}=-1+\frac{1}{1+x^2}$$
so the given sum which vanish at $0$ is
$$\int_0^x\left(-1+\frac{1}{1+t^2}\right)dt=-x+\arctan x$$
