How can I solve the integral in the error function $\mbox{erf}(x)$?

Question

To get from this

To this series

I can't seem find the step-by-step solution anywhere.

Answer 1

Note that \begin{align*} \int e^{-x^2}\,dx &= \int\left\{ \sum_{k=0}^\infty \frac{1}{k!}\left(-x^2\right)^{k} \right\}\,dx \\ &= \int \left\{ \sum_{k=0}^\infty \frac{1}{k!}(-1)^kx^{2k} \right\}\,dx \\ &= \sum_{k=0}^\infty(-1)^k\frac{1}{k!} \left\{ \int x^{2k}\,dx \right\} \\ &= \sum_{k=0}^\infty (-1)^k\frac{1}{k!}\frac{1}{2k+1}x^{2k+1} \end{align*}