Sum of power series

Question

Good morning, I have difficulties to find an approximation formula (or bound from the height) for the sum of the following power series $\sum \limits_{k=1}^\infty e^{-k^2}x^k$. Thanks

Answer 1

You can use the Euler-MacLaurin summation formula in the following way:

$$\sum_{k=1}^\infty e^{-k^2}x^k\sim \int_1^\infty x^y e^{-y^2}dy+\frac{1}{2}e^{-1}x=-\frac{\sqrt{\pi }}{2} e^{\frac{\log^2x}{4}} \left(-1+\text{Erf}\left[1-\frac{\log x}{2}\right]\right)+\frac{1}{2}e^{-1}x$$

This gives an asymptotic estimation to this series.