How can we find the values that a (divergent!) series tends to? Suppose we are given a series that diverges.  That's right, diverges.  We may interest ourselves in the limiting function(s) of its behavior.
For instance, given the power series:$$\frac{1}{1+x} = 1 - x + x^2 - x^3 + \dots$$
I am interested in finding the sum of the coefficients for $x^n$ and $x^{n+1}$ as $n$ approaches infinity.  This should be fairly obvious as to what it is, but is there some way that we could do this for a general alternating series that essentially converges to two values, or more importantly, for 3 or more values (i.e. not exactly an alternating series, but one that cycles through a set of values at infinity)?
MY IDEAS
I guess that this can somehow be accomplished similar to finding the limiting behavior for a convergent series.  I thought that I knew how to do this, but I forgot what I thought was an easy way.  
MY GOAL
I really would like to know if it's possible to apply a function to the values at the limit.  If it is, of course, I'd like to know how to do this.  This may allow us to determine what the values actually are by using multiple functions.

Note that the usual definition of the infinite sum is a very different kind of thing from an ordinary finite sum. It introduces ideas from topology, analysis or metric spaces which aren't present in the original definition of sum. So when generalising from finite sums to infinite sums we have quite a bit of choice in how these concepts are introduced and it's something of a prejudice to call the standard method from analysis the sum.
There are quite a few different approaches to summing infinite series, many of which give finite values in places where the usual summation method fails. Examples are Abel summation, Borel summation and Cesàro summation. The mathematician Hardy wrote an entire book on techniques to sum divergent series.
These alternative summation methods aren't just a mathematical curiosity. They play a role in physics where they can help sum some of the series that arise from sums over Feynman diagrams. These results often agree with results computed by more orthodox methods.
