How can I make a series expansion of $F(x) = \int_0^x \exp -{(t^2)}\ dt$? $$F(x) = \int_0^x \exp -{(t^2)}\ dt$$
We need to find the series expansion for $F(x)$.
I tried differentiating $F(x)$ but couldn't establish certain pattern so that Taylor series formation may help..
Kindly help !!

You know the expansion for $\exp(t)$, hence the expansion for $\exp(-t^2/2)$. Plug in, integrate, and by happy: $$\exp(t) = \sum_{n \geq 0} \frac{t^n}{n!} \implies \exp\left(-\frac{t^2}{2}\right) = \sum_{n \geq 0}(-1)^n \frac{t^{2n}}{2^nn!},$$ so we get: $$F(x) = \sum_{n \geq 0}(-1)^n\frac{x^{2n+1}}{(2n+1)2^nn!}.$$
Edit: for the edited question with $\exp(-t^2)$ the same reasoning works. It is in fact easier.
