On summation of the Taylor series of the function 1/(1-z) by the theta summation method

The family of the Taylor series f_{\epsilon}(z)= \sum\limits_{0\leq{}n<\infty}e^{-\epsilon n^2}z^n is considered, where the parameter \epsilon, which enumerates the family, runs over ]0,\infty[. The limiting behavior of this family is studied as \epsilon\to+0.