Finding the taylor series with center of zo=0

Question

how to Find the taylor series with center of zo=0 and convergent region for $f(x) = \int_{0}^{x} e^{-t^2}\,dt $.

Answer 1

Since

$$ e^t = \sum_{n \geq 0} \frac{ t^n }{n!} \implies e^{-t^2} = \sum_{n \geq 0} \frac{ (-1)^n t^{2n} }{n!}$$

Thus,

$$ f(x) = \int\limits_0^x \sum_{n \geq 0} \frac{ (-1)^n t^{2n} }{n!} dt = \sum_{n \geq 0} \frac{ (-1)^n x^{2n+1} }{(2n+1)n!} $$

Answer 2

We have $f(0)=0$ and $f'(x)=e^{-x^2}$

Now its your turn.

Answer 3

For an alternative derivation, do you know the series for $e^x$? How can we turn that into a series for $e^{-x^2}$? Do you know how we can integrate power series?