summation of $e^{(f(x))}$

Question

$$\sum _{x=0}^4e^{-\left[0.06\left(\frac{2x+1}{2}\right)+0.001\left(\frac{2x+1}{2}\right)^2\right]}$$

Hi, I am new, I am not sure if the above code works here, but I want to find

$$\sum_{x={0}}^ 4 [ exp(-{0.06(2x+1)/2 + 0.001[(2x+1)/2]^2}) ]$$

I know how to sum e^(-x) as it can be treated with as a geometric series. I want to know how to find sum of e^(f(x)) in general.This question is for technology access, but I also want to know if I can simplify it by hand first so that scientific calculators can compute it. Thank you.

Answer 1

A series of the form $\sum_{n=0}^{\infty} e^{-(an^2+bn+c)} $ is a Theta function.

Here is a typical source of information:

https://en.wikipedia.org/wiki/Theta_function

There is an incredible amount of information about Theta functions.

Here is an inexpensive introduction:

https://www.amazon.com/Brief-Introduction-Theta-Functions-Mathematics-ebook/dp/B00GU6GOBM/ref=sr_1_1?ie=UTF8&qid=1492233768&sr=8-1&keywords=bellman+theta+functions

A warning: There is so much information about the various types and definitions of Theta functions, that it is almost as bad as tv tropes.