Consider the following result（$d$ denotes the dimensions and $0<t<T$） $$c\left(\sum_{j=0}^\infty\frac{\Gamma^j(1-\kappa)}{\Gamma((j+1)(1-\kappa))}t^{j(1-\kappa)-\kappa}\right)^{\frac{1}{2}}\leq c t^{-\frac{\kappa}{2}}, \text{ where } \frac{\kappa}{2}=\frac{d}{\alpha}-\frac{d}{2q\alpha}, \alpha+d\geq \frac{\alpha+d}{2q} ，0<\alpha\leq 2， \text{ and } 2q>1,$$ which appears in the article Quantitative normal approximations for the stochastic fractional heat equation. I notice that it only takes the term of $j=0$. I try to use the conclusion $\Gamma(z)\sim z^{z-1/2}e^{-z}$, and prove it by considering the rapid decrease. But I don't know if it is true and how to explain it convincingly.

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