On first blush, it might seem that if you add an infinite number of positive numbers together, the sum must be unbounded. But if the numbers are getting small enough fast enough, the sum of an infinite series can be both finite and exactly defined.

Probably the simplest infinite sum is the powers of ½. Let's say that half of a room is painted in the first hour, and half of what remains is painted in the second hour, so now 3/4 of the room is painted. For reasons unknown, you decide to let this increasingly lazy person continue their plan of painting less and less each hour, one eighth in the third hour, one sixteenth in the fourth hour, on and on infinitely. I say infinitely because mathematically the room is never finished. (In reality, we will get to a sliver so tiny that it must be painted completely, if only because a molecule of paint once split isn't a molecule of paint anymore.)

In math, this is well understood now to be an infinite sum that adds up to 1. You might fairly say "we never get there", which is true. This was the point of one of the famous paradoxes of the ancient Greek mathematician and philosopher Zeno. But in modern math we have the concept of a limit as

*n*approaches infinity. We can agree that the sum never gets to be more than 1, and the idea of a limit is that you get to choose how close you want the sum to be to 1 and I have to give a number of steps which will guarantee a sum that is closer than your given definition of "close enough". For example, if we want the sum to be within one millionth of 1, which is to say larger than .999999 but still less than 1, I can guarantee that after twenty steps, the sum will be that large, because ½ raised to the twentieth power is slightly less than one millionth.

Tomorrow, one of the tricks used for the general power series of positive numbers less than 1.