### Messing around with infinity

Infinity is a fascinating topic. When we start taking a close look at infinity (as if this was possible) we realize that infinity is actually something much more tangible than one may think. First, it is amazing to see that some infinite sums actually converge. Counterintuitive, isn't it? Starting with something like this: 1 + 1/2 + 1/4 + 1/8 +.... There is a very simple technique to solve this problem. Here it is: S = 1 + 1/2 + 1/4 + 1/8 +.... Now multiply both sides by 2 and you get: 2S = 2 + 1 + 1/2 + 1/4 + .... Take a look closely at this : 2S = 2 + 1 + 1/2 + 1/4 + .... Well, this is S, right? Hence: 2S = 2 + S which leads to S = 2 . One way to think about the above sum is as follow: suppose that you want to cross a bridge that is a mile long, and you want to cross it and come back. So you cross it first, travelling 1 mile. On your way back, first you have to cross half of the bridge (1/2). Then the half of the half (1/4). And so on. Hence the distance that you