After reading answers (especially M. Piau's) to my other question Asymptotic behaviour of some series, the more general question came to my mind, namely:
how to examine the asymptotic behaviour as $x\to 0^+$ for the following series:
$$\sum_{n=1}^\infty\sin^2\left(\frac{n\pi}{a}\right)\exp\left(-\frac{n^2\pi^2 x}{4}\right),$$ where $a>0$ is arbitrary, i.e. how to find a simple continuous function $g$ such that the above series is equivalent to $g(x)$ as $x$ goes to $0^+$.
I suspect that the answer would be again related to $\mathrm{const}\dfrac1{\sqrt x}$ but I just couldn't show it by using the approach of M. Piau. The problem now is that $\sin^2\left(\dfrac{n\pi}{a}\right)$ could take even infinitely many different values between $0$ and $1$. Once again thank you for any replies.
Asymptotic behaviour of some series II
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real-analysis
sequences-and-series