how do you find the sum of an infinite series?
Depending on the series, there are tricks sometimes ... often not. A well-known case are the geometric series where you can multiply by a factor of (x-1) and everything cancel in a nice way. It only converges for $\displaystyle 1 \geq r \geq -1 $ though.
$\displaystyle \sum_{i=0}^n x^i = \frac{1-x^{n+1}}{1-x} $
Mr. F is right, this question is WAY to broad to answer...this goes right up there with "How do you integrate?" and "How does one drive a vehicle?"...but Im sure youll come back and give us an example
I do think I ought to mention some of the most common ways. They include (but of course are not limited to): power series, fourier Series, riemann sums, telescoping sums, and function definitions.
This of course is itself a very broad and unfufilling answer, but this is intentional. Research these methods and if you have any questions pertaining to them or others just ask. Im sure either myself or one of the other members here will be glad to oblige your hungry mind
EDIT: Jeez...I sound kinda loopy and eccentric dont I