Infinite Series

Abstract

At one time or another nearly everyone has seen the problem which proposes that although we start toward some place, we never get there. The reasoning, of course, is that we first go half way, then we go half the remaining distance, namely one fourth of the way, and after that half the remaining distance, that is one eighth of the way, etc. We can get close enough for almost any intended purpose, but we cannot actually arrive. This is equivalent to considering the non-ending sum 1/2 + 1/4 + 1/8 + 1/16 + ⋯. From our work in Chapter I we realize that this is a geometric progression, and therefore the sum of the first n term is

$${{{{\left( {1/2} \right)}^{n + 1}} - 1/2} \over {\left( {1/2} \right) - 1}} = 1 - {\left( {1/2} \right)^n}$$

. We observe that the distance that separates us from our goal is (1/2)ⁿ, and after a sufficient number of terms this will be small enough to ignore, whatever that may mean. We are tempted to say, then, that this sum of an infinite number of terms has the value one.