Anyone who's taken a beginning calculus course spent some time studying (and maybe dreading) infinite series.  A convergence definition was probably given as the convergence of the sequence of partial sums of the series.  Next the course probably talked about various ways to tell when a series converges or diverges.  From then on, very little time was spent on series that diverge, as they're clearly not very interesting.

And, oh what a lie that is.  Divergent series have held mathematicians interest for quite some while.  If you were lucky your calculus class might've even spent some time talking about the divergent harmonic series 1 + 1/2 + 1/3 + .... This series diverges, however it diverges in a very interesting way.  The sequence of partial sums of the harmonic series closely approximate the natural logarithm function.  In fact, if H_n is the nth partial sum of the harmonic series, lim(n -> infinity) (H_n - ln n) is a constant .57721566... 

In general mathematicians are interested in finding ways of making divergent series converge.  As a motivating example, consider the series 1 -1 + 1 -1 + 1 - ...  The partial sums of this series are 1, 0, 1, 0, ...  These partial sums oscillate back and forth between 1 and 0, so this series clearly doesn't converge to anything.  However, an interesting property of this series is that on the odd chance that it did converge, we know what it would converge too!  To see this, let S = 1 - 1 + 1 -1 + 1... Then S = 1 - (1 - 1 + 1 - 1 + 1 ...) = 1 -S, so S = 1/2.  This gives us a great test.  If we can find any method of making this divergent series converge, this method must cause the series to converge to 1/2.

Now there's no hope of making the series converge using the Cauchy definition of convergence that we were taught in our calculus course.  We need to search for other ways of defining convergent series.  While we search for alternate definitions, we shall insist on one thing.  Any series that is convergent using the calculus-course definition of convergence must also converge using our new definition, and furthermore must converge to the same value.  This means that our new convergence techniques will expand our universe of convergent series, however it won't change anything we already know.

For the alternating series above, let's start by considering the alternating geometric series 1 -x + x^2 -x^3 + x^4 - ...  As long as |x| < 1, we know that this sums up to 1/(1+x).  If we let x get closer and closer to 1 (while remaining smaller than one) the series will sum to a result closer and closer to 1/2.  If x = .999, the series sums to .50025.  If x = .9999, the series sums to .500025, and so on.  This gives us another way to think about our divergent alternating series above.  Why don't we consider all of these convergent geometric series where |x| < 1, and take the limit as x approaches 1?  Instead of taking limits of more and more terms in the series, where taking limits as the terms of the series approach a number.  This gives us 1/2 as the sum of the series, and motivates the following definition:

Definition:  Let a0 + a1 + a2 + ... be a series, and assume that there is a function f with a power series a0 + a1x + a2x^2 + ... convergent for -1 < x < 1.  If f(x) is defined and continuous at x = 1, say that a0 + a1 + a2 + ... converges in the sense of Abel, and that the sum converges to f(1)

A few minutes of thought should convince you that any series that converges in the usual sense also converges in this Abel sense, and it converges to the same value.

Another way of defining convergence is by the use of Cesaro sums.  The series 1 -1 + 1 -1 +... diverges not because the partial sums grow uncontrollably, but rather because the partial sums oscillate around too fast.  If we could find a way of averaging the sums in order to smooth them out, maybe this series will converge.  And of course we can do exactly that. 

Given a series a0 + a1 + a2 + ..., let Si be the ith partial sum of the series.  So S0 = a0, S1 = a0 + a1, and so on.  The nth Cesaro sum is defined as (S0 + S1 + S2 + ... + Sn)/(n+1), the average value of the first n+1 partial sums.  if the Cesaro sums converge to a finite value as n grows to infinity, we say that the series a0 + a1 + a2 +... converges in the sense of Cesaro, and that the sum converges to the limit of the Cesaro sums.

Now in the case of the series 1 -1 + 1 -1 + ..., the Cesaro sums converge to 1/2.  Take a minute to convince yourself that if a series is convergent in the normal sense, the Cesaro sums converge to the normal sum of the series.  It's not quite as simple as it was with Abel sums, however it's not too hard.

These are two of the most basic alternative convergence definitions out there.  There are a number of others that contribute to this interesting field.  This can also explain some surprising results, such as Euler's assertion that 1 + 2 + 3 + 4 + ... = -1/12.