A Geometric Proof of an Infinite Series

In the previous post, we have seen how easy it is to prove a problem in Geometry using Algebra. The problem could also be proven geometrically, but the proof is longer. In this post, we will learn how to use Geometry to prove a problem on infinite series. That is, we have to show geometrically, the sum of

\displaystyle\frac{1}{2} + \displaystyle\frac{1}{4} + \displaystyle\frac{1}{8} + \displaystyle\frac{1}{16} + \displaystyle\frac{1}{32} + \displaystyle\cdots.

Note that the symbol \cdots means that the number of terms is infinite; that is, the addition continues without end.  In this post, we show that the sum of this infinite series is 1.

Theorem

\displaystyle\frac{1}{2} + \displaystyle\frac{1}{4} + \displaystyle\frac{1}{8} + \displaystyle\frac{1}{16} + \displaystyle\frac{1}{32} + \displaystyle\cdots = 1

Proof

Notice that each term after \frac{1}{2} is half of the previous term. If we represent each term with a part of a square with area 1 square unit, the sum the series is equal to the sum of all the partitions as shown below. Clearly, now matter how many times we partition the square in relation to the series, it will never exceed 1. In fact, as we partition, the sum of the areas gets closer and closer to 1. Since, we divide indefinitely, the sum of the areas is equal to 1.

infinite-series

Therefore,

\displaystyle\frac{1}{2} + \displaystyle\frac{1}{4} + \displaystyle\frac{1}{8} + \displaystyle\frac{1}{16} + \displaystyle\frac{1}{32} + \displaystyle\cdots = 1. \blacksquare

In the proof above, we can see that a geometric figure can be used to prove a number problem.  Proofs such as this that requires no explanation to show that a particular theorem is true is called proof without words.

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