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
$\pi$.

It depends on the function.  In your first question you have the generating function for central binomial coefficients and a variety of techniques such as those described in generatingfunctionology.  Your equation appears as (2.5.11)
In your second question, it is unclear exactly what you are asking, as the first equation is for me the definition of $e$.  
Note that $\sum_{n=0}^\infty \frac{x^n}{n!}$ is its own derivative and so a multiple of $\exp (x)$, and looking at $x=0$ it must be $\exp (x)$, so back to your question with $x=1$ the value is $\exp(1)$ or $e$.  
Or perhaps you want an approximation to the decimal value: for example the terms up to $n=9$ add up to 2.7182815... and it is easy to show that the others add up to less than 0.000001 so the sum is about 2.71828... 
