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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

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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.

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    The series $\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}\left(f^{2k-1}(\infty)-f^{2k-1}(1)\right)$ is negligible? Thanks.2012-02-14
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    I have written just the leading order of an asymptotic expansion. You can account for all the series by considering also these terms.2012-02-14
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    Pardon my bad English, but I meant to ask "why the previous series is negligible"? Sorry.2012-02-15
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    @Mark: These should be considered as higher order terms into an asymptotic series. In your case you will get the correction $$\frac{1}{32}e^{-1}x\ln(x/e^2)$$ and powers of this that I have neglected but that you can consider depending on the numerical precision you may need.2012-02-15