What is the sum of infinite numbers?

Answer 1

The sum of the infinite is infinite or a finite number, depending on the numbers that you are summing up.

Sometimes an infinite series will converge to a finite answer. An example of one that results in an infinite answer should be fairly easy. Consider 1+2+3+4+5+6+.... Each number is bigger than the previous.

But what about when each term is smaller than the previous. Consider this example, which most people should be familiar with. Take the decimal equivalent for 1/3, which is 0.3333333.... We know this is a finite number. This can be written as an infinite series 3/10 + 3/100 + 3/1000 + . . . . + 3/(10n). We would say that this infinite series converges to 1/3.

Look at this one: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .... Each term is 1/2 the previous term. As the terms are added, the sum of the series would look like this: 1/2, 3/4, 7/8, 15/16, 31/32,... Notice that each sum is half way between the previous sum and 1, but will never get to 1. This series converges to 1.

Not every series, where the terms decrease, will converge to a finite number though. I won't show how, here, but the series 1/2 + 1/3 + 1/4 + 1/5 + . . . + 1/n, does not converge but goes to infinity. Each term is smaller than the previous, but they are not getting small 'fast enough' to converge to a finite number.