Question

Why can a series wich is condiontally converging be rearanged to converge towards any chosen value?

Answer 1

Both positive and negative diverge on their own. That's the crucial part. Suppose I have 1+1+1+... and -1-1-1..., I can rearrange it to be "more positive" by doing 1+(1-1)+(1-1)+... to get 1, or 1+1+(1-1)+(1-1)+... to get 2, or 1+1+...{k times}+(1-1)+(1-1)... to get arbitrary positive k. Of course this is not an example of conditionally convergent series but it illustrates the point of what you can do when each positive and negative parts diverge.

Answer 2

Have you actually tried it with an example? Like, take a nice conditionally convergent series like 1-1/2+1/3-1/4+1/5-... and make it converge to pi by rearranging. Literally take a calculator and add positive summands one by one until you're above pi. Then add negative summands one by one until you're below again. Then positive summands, and so on. Try it out, you may get a better feeling for why it works.