How to find what a given series/sequence converges to 
*

*Given a function series, what are
the ways often used to find the
function that the series converges
to?
For example, how do you construct
the function on the right hand side
from the series on the left hand
side: $$\sum_{n \in \mathbb{N} \cup \{ 0\}} (\begin{array}{c} 2n \\ n
    \end{array}) x^{n}=(1-4x)^{-1/2}$$ I suppose one
may try to go the other way around,
i.e. verify that the Taylor
expansion of the function on RHS around 0 is
the series on the LHS, but I was
wondering if this is one of the
acceptable way to construct the RHS
from the LHS?

*Related questions:
Given a sequence of real numbers,
what are the ways to find the number
the sequence converges to? 
What are the ways to find the number
a given series converges to? For
example,
$$\sum_{n=0}^\infty \frac{1}{n!}=e$$
and the series that converges to
$\pi$.

In general most power series and most sequences do not have closed forms. The specific series you mention is hypergeometric and there are known algorithms for working with these, e.g. guessing or proving identities or algebraic or differential equations. I don't know much about this subject, but Petkovsek, Wilf, and Zeilberger might be a good place to start. (If you just want to get from the RHS to the LHS, use the generalized binomial theorem.)
I do not understand your second example. $\sum \frac{1}{n!} = e$ is more or less a definition. If you define $e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n$, then you can prove this by proving that $e^x = \sum \frac{x^n}{n!} = \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$, which is a nice exercise. 
There are special techniques which work on series that have a particular form, but again, there is no reason to expect that an arbitrary sum has a reasonable expression in terms of known constants. 
