let $a_n = \dfrac{e^{-(1/2) \times a^2 \times\log(n) }}{a\sqrt{2\pi \log(n)}} $, $a$ is a constant, and the question is if
$S_n = \sum a_n$ converge to a finite number.
I wonder if I should use integration to do it (cauchy's method)? and if the result depends on the value of $a$?
I just run some code and find that when $a > \sqrt{2} \rightarrow{}$convergence, otherwise divergence, anyone can provide some test to justified it theoretically?
$e^{-\frac{1}{2}a^2\log(n)}=n^{-\frac{1}{2}a^2}$ and $$ \frac{e^{-\frac{1}{2}a^2\log(n)}}{a\sqrt{2\pi \log(n)}}\ge k\frac{n^{-\frac{1}{2}a^2}}{\log(n)}$$ for $n\ge 3$. where $k=\frac{1}{a\sqrt{2\pi}}$.
let $a\le \sqrt{2}$. then $$ \ge k\frac{n^{-1}}{\log(n)}$$ and $$ \sum^{\infty}_{n=3}\frac{1}{n\log(n)}$$ is divergent series. if $a>\sqrt{2}$, $$ \frac{e^{-\frac{1}{2}a^2\log(n)}}{a\sqrt{2\pi \log(n)}}\le k\frac{1}{n^{\frac{1}{2}a^2}}$$ and $$ \sum\frac{1}{n^{\frac{1}{2}a^2}}$$ is convergent series.