Sum of Gaussians

Q: I would like to find a closed form for the sum

Clearly is periodic with unit period (put and ). How can I compute the Fourier series expansion of ?

A: Putting and into the formula functions.wolfram.com/09.03.06.0019.01,

that is,

we can express the sum in terms of EllipticTheta as follows:

The Fourier series expansion of (essentially its standard definition) is given at functions.wolfram.com/09.03.06.0001.01,

Here are the first few terms of the Fourier series expansion of .

Note that the Fourier coefficients decrease extremely rapidly.

An excellent approximation is obtained by keeping just the first term in the sum, that is, .