OK, so my mind has officially been blown. In a completely good way, however.
Everyone knows basic properties of algebra. We know for example, that whether you write out 2X3 or 3X2 you will still get a product of 6. We know that whether you write 3 + 2 or 2 + 3 you will still get a sum of five.... or will you?!? mwahaha not always!
What? Yeah, that's what I said too. You might be shocked to find that when dealing with sums of an infinite number of terms, commutitivity of algebra doesn't hold. This means that, you
will get a different sum based simply on the
order of the terms you are adding.
When dealing with infinite series, one is forced to deal with this problem. For some background, an infinite series generated by some sequence X is itself a sequence, where partial sums are taken for each element in the sequence.
To make matters easier, it is simply a summation, like what is pictured in the graphic in the next paragraph to the left.
Consider a summation of the form:
1 - 1 + 1/2 - 1/2 + 1/3 - 1/3....
this is an alternating harmonic series, a very common series to come across in mathematics. This sequence converges quite obviously to 0, however that convergence is conditional, rather than absolute convergence.
You will see a simple rearrangement of terms, perhaps:
1 - 1 + 1/2 + 1/4 - 1/3 + 1/5 + 1/6 + 1/7 + 1/8 - 1/4.....
will not converge to 0 at all, in fact it will diverge to positive infinity.
Think about it... every term is still represented. 1/2 is there, - 1/2 is there, 1/4 is there as well as 1/80000 and -1/80000, however they are just ordered differently.
In fact, any conditionally convergent series (meaning that a rearrangement can change the limit of the sequence) has an interesting property. For any number L, there exists some rearrangement of terms such that you can force the series to converge to that number.
So there you go. I hope it blew your mind as much as mine.
