What IS conditional convergence? I've gone through countless websites that promise answering "What is conditional convergence?" and instead give me "This is how you find if something is conditionally convergent". 
While it is all fine and dandy that I know how to find if something is conditional convergence, what actually is it?
I've graphed severally conditionally convergent series and they seem to not look much different from other series which have absolute convergence. What is the magic thing that makes something converge conditionally? I'm just confused because I don't understand how something can converge but actually not really all the time.
 A: You can find interesting Sequences and Series: A Sourcebook by Pete L. Clark. In elementary real analysis the following two conditions are equivalent:

*

*$\sum |a_n|$ is convergent (absolute convergence).

*$\sum a_n$ is convergent and every rearrangement converges to the same sum (unconditional convergence).

But "...in functional analysis one studies convergence and absolute convergence of series in a more
general context, such that nonabsolute converge and conditional convergence may indeed differ." (p. 62)
A: "Conditional" is a bit of a strange adjective to use. After all, a series either converges or it doesn't: what is conditional about that?
The reason for the word "conditional" is that, given any series which converges but does not converge absolutely, it is possible to rearrange the series (i.e., reorder the terms) in such a way that the series no longer converges.
It is also possible, given any desired value $V$, to find a rearrangement of the series which converges to $V$.
This is known as the Riemann rearrangement theorem.
Note that this phenomenon does not occur with absolutely convergent series. Given any absolutely convergent series, we can rearrange the terms any way we like, and it will still converge to the same value.
A: The definition is in other answers, so here I just add some explaination to make it more understandable.
One of the purpose of such definition (converges conditionally, converges absolutely) is to provide a nice way to decide whether rearrangement of a series will cause some problem. Consider the series:
$$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac12 + \frac13 - \frac14 + \cdots$$
It converges, but if we rearrange these terms in some particular order, it will converges to a different limit. What we want here is to decide when and why the commutativity in finite setting is preserved in infinite setting. With the help of these definitions we can say:

*

*If a series converges absolutely, then any rearrangement of this series converges to the same limit. (this is a nice quality to have since rearrangement has no "side effect" here)

*If a series converges conditionally, then we need to worry about rearrangement. (warning: rearrangement may cause it converges to different value!)

The magic thing that makes something converges conditionally is the interaction between negative and positive terms, and there's no bound on the sums of either positive terms or negative terms. So with infinite many negative terms and positive terms in hand, you can let a conditionally convergent series converges to any value you want by rearrange the terms in a proper way.
