Did you know...?

[Alex K]

Wow cool

I've not heard that one before. So do you mean adding the terms up in a different order?

Yes, a simple example of an infinite permutation is this: add them by always taking the next two even ones and then one of the odd ones, then again two even ones, then one odd, you get the idea:

1/2 + 1/4 - 1/3 + 1/6 + 1/8 - 1/5 + ....

This series will eventually reach each term of the original one, but it converges to roughly 0.65 instead of 0.31 ! This always happens with series which are only convergent if you have alternating signs, but will diverge if you put "+" everywhere as in this case. Any series which is not still convergent if you put "+"es (is not "absolute convergent"), can be rearranged to yield an arbitrary real number. It's a theorem, I forgot how it is called.