## Bounds on Gaussian infinite sum

What are some good upper and lower bounds on the following sum? S=+∞∑n=−∞1σ√2πe−12(nσ)2 I am looking for something better than 1<S<2. Answer If fact, there is an explicit solution to the expression S=+∞∑n=−∞1σ√2πe−12(nσ)2=1σ√2πϑ3(0,e−12σ2) where appears the elliptic theta function. The function decreases asymptotically to 1 but it goes to infinity for small values of σ. … Read more