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
$$\sum\limits_{i=1}^\infty  (-1)^i\frac{x^{2i+1}}{2i+1}$$

Hint: Expand $\frac{1}{1+t^2}$ using the familiar $\frac{1}{1-y}=1+y+y^2+y^3+\cdots$, and integrate term by term from $0$ to $x$.  Then adjust for the fact your sum starts at $i=1$.
