# Infinite sum: $\sum_{n=-\infty}^{\infty} \frac{1}{10^{{(n/100)}^2}}$

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?

• What leads you to believe that there is a nice closed form that doesn't involve $\vartheta$? – Mark Viola Nov 26 '16 at 0:45
• @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. – Nico A Nov 26 '16 at 1:50

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 $$3 \left(5 (5+\pi )-\sqrt{\pi }\right)\approx116.8065282511303504950451$$