The question asks to find a series expansion of $\int_{-\infty}^{x}e^{\frac{-s^2}{2}}ds$, given: $$\int_{-\infty}^{x} \frac{1}{s^n}e^{\frac{-s^2}{2}}ds=\frac{-e^{\frac{-x^2}{2}}}{x^{n+1}}-(n+1)\int_{-\infty}^{x} \frac{1}{s^{n+2}}e^{\frac{-s^2}{2}}ds$$ I've come with with this series: $$I(x)=\sum_{j=0}^{\infty}(-1)^j \frac{1}{x^{2j+1}}e^{\frac{-x^2}{2}} \prod_{k=0}^{j} (2k+1)$$
but the ratio test shows that this series is divergent, which I think shouldn't be the case as the integral itself converges for all $x$.
edit: not a Taylor expansion! I'm required to come up with a series representation with the given recursive relation.
Any help would be greatly appreciated!