Rearranging terms of a conditionally convergent series

[Nikitin]

Why the heck will the sum of such a series going towards infinity change if its terms are re-arranged? My book omits the proof, and without it this claim makes no sense to me.

Can somebody provide an example of such a series, and maybe some light explanation (I'm way too exhausted for heavy stuff)?

 

[mfb]

Wikipedia has examples
The basic idea is to shift many elements with one specific sign away ("to the right") to get the sum to diverge in the chosen direction.

 

[Robert1986]

The idea is something like this: pick your favorite number, L. Add up some positive terms until you get something that is bigger than L. Then add some negative terms until you get something that is less than L, then add some positives until you get bigger than L, and so on. Since the series is not absolutley congergent, you will always be able to do this.