Series are fun.

We learned in grade school that order in which one adds a bunch of numbers doesn't matter. So . This is not necessarily true when you are adding infinitely many numbers. Consider this series: and this one: The question is: are they equal? No! Here's how to show it. Let be the -th partial sum of , and be the same thing for . Let . These are finite sums, so we can move the numbers we add around. In particular, let's separate them into two groups of additions and subtractions: and similarly: Take the limit : For absolutely convergence series, the sum is not affected by changing the order.