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I need help with the following infinite sum: $$\sum_{n=-\infty}^{\infty} \frac{1}{10^{{(n/100)}^2}}$$

It can quite obviously be expressed in terms of the Jacobi Theta Function, but I feel like that is more of a "re-definition" than an actual closed form. There are a few elementary closed forms that may work, but I can't get a very accurate numerical estimate. The approximate numberical value is $\approx 116.589$

How can I derive a closed form for this sum?

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  • $\begingroup$ What leads you to believe that there is a nice closed form that doesn't involve $\vartheta$? $\endgroup$
    – Mark Viola
    Nov 26, 2016 at 0:45
  • $\begingroup$ @Dr.MV Because a lot of elementary closed forms seem to come really close, or may even be equal (most prominently $100 \sqrt{\frac{\pi}{Log(10)}}$), and the fact that the summand is so simple. $\endgroup$
    – Nico A
    Nov 26, 2016 at 1:50

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As you wrote,$$\sum_{n=-\infty}^{\infty} \frac{1}{10^{{(n/100)}^2}}=\vartheta _3\left(0,\frac{1}{\sqrt[10000]{10}}\right)\approx 116.80652181457340815470396$$ According to advanced inverse symbolic calculators, the closest expression is the one you give in comments $$100 \sqrt{\frac{\pi}{\log(10)}}\approx 116.80652181457340815470400$$ and nothing better can be found (at least by myself).

Not "too bad" is also $$3 \left(5 (5+\pi )-\sqrt{\pi }\right)\approx116.8065282511303504950451$$

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  • $\begingroup$ Thanks for the answer! Do you have any idea how I would go about proving any of these? $\endgroup$
    – Nico A
    Nov 26, 2016 at 13:54
  • $\begingroup$ No idea ! Sorry for that. Cheers :-) $\endgroup$
    – Claude Leibovici
    Nov 26, 2016 at 15:17

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