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I have been trying to get a lower bound on the following alternating sum but without much success:

$$ \sum_{j=1}^T (-1)^j e^{-j^2} j^k . $$ For small values of $k$, this is easy because the first term dominates the remaining terms.

For an upper bound, I can get an estimate using the gamma function by ignoring the sign change. (Basically this reduces to estimating the moments of some Gaussian distribution.)

In general, what I want to estimate are summations of the form

$$ 2^{-n} \sum_{j>0} \binom{n}{n/2 + jM} (-1)^j (jM)^k . $$

(It may be useful to think of $M = C\sqrt{n}$ and $k = 1$ at first, but my goal is to understand the sum for other values of $M$ and $k$ in general.)

Is there any general method for estimating this kind of alternating sums? Thanks.

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    $\begingroup$ Since $e^{-j^2}j^k$ is decreasing eventually, we're in the situation of the Leibniz criterion and obtain bounds from this (though perhaps not of the kind you were hoping for). $\endgroup$
    – Christian Remling
    Commented Sep 23, 2020 at 1:15
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    $\begingroup$ Use Abel’s bound and Abel Summation By Parts and then $\sum_{j=1}^k j^k$ related to Bernoulli numbers google.com/url?sa=t&source=web&rct=j&url=http://… $\endgroup$
    – user160903
    Commented Sep 23, 2020 at 6:27
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    $\begingroup$ Let $(a_n)_{n\in N^*}$ be a sequence of Real or Complex numbers and $\varphi$ a Real or Complex function of class $C^1$.Let $A(x) = \sum_{1\le n \le x}{a_n}.$, then for all $x$ Real ,$\sum_{1\le n\le x}a_n\varphi(n)=A(x)\varphi(x)-\int_1^xA(u)\varphi'(u)\,\mathrm du$ $\endgroup$
    – user160903
    Commented Sep 23, 2020 at 6:45
  • $\begingroup$ Take $a_j=j^k$ and $\varphi(j)=(-1)^j e^{-j^2}$ $\endgroup$
    – user160903
    Commented Sep 23, 2020 at 6:50
  • $\begingroup$ I think you Can find an approximation formula of your sum by using Abraham de Moivre formula for $\log n!$ $\endgroup$
    – user160903
    Commented Sep 23, 2020 at 7:34

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