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I'm wondering how to examine the asymptotic behaviour as $x$ goes to $0^{+}$ for the following series (which is the continuous function in the right neighbourhood of $0$):
$$\sum_{n=1}^{\infty}\sin^{2}\left(\frac{n\pi}{2}\right)\exp\left(-\frac{n^2\pi^2 x}{4}\right)$$ i.e. how to find a simple continuous function $g$ such that the above series is equivalent to $g(x)$ as $x$ goes to $0^{+}$. Thank you for any replies!

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    What do you think of $g(x)=\frac{1}{e^{\frac{\pi^2x}{4}}-1}=\sum_{n=1}^{\infty} \exp\left(\frac{- n \pi^2 x}{4}\right)?$2011-11-12
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    Nice function, but in a view of other answers it looks like it's not correct.2011-11-13

4 Answers 4