A proof without words is a proof of a mathematical statement which can be demonstrated as self-evident by a diagram without using explanatory texts. These visual proofs are considered more elegant than the formal proofs. As a student, you must have experienced that the visual memory is more powerful than the linear memory, where we memorize the sequence of various mathematical steps. We read and write all these books on mathematics and science to make sense of a visual world. So, wouldn’t it be great if we can understand these concepts visually, in the first place itself. In this post, we will discuss one of the most famous proof without words, which is of the sum of the geometric series,

1/2 + 1/4 + 1/8 + … = 1

sum of infinite geometric series, 1 over 2 to the power N, where N goes from 1 to infinity
sum of infinite series

This series is geometric, because each of the successive term can be obtained by multiplying the previous term by 1/2. Geometric series have played an important role in the development of early calculus, and are central in the study of convergence of series.

The Visual Proof

We start with a 1X1 square with area 1. Our first term is 1/2. So we color half of it in blue. The white represents the left out area, which is again 1/2.

First term of 1/2 + 1/4 + 1/8 + ...
First term of 1/2 + 1/4 + 1/8 + …

The next term 1/4 will be half of the white, and we color it green.

sum of first 2 terms
1/2 + 1/4

The next term 1/8, is half of the white, and we color it red.

sum of first 3 terms
1/2 + 1/4 +1/8

We keep coloring the area for the next term, each of which is half of the remaining white area.

proof without words - 1/2 + 1/4 + 1/8 + 1/16 + ...
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256

What we notice is, no matter how small the remaining white area is, it is big enough to contain all of the next terms in this infinite series. You can watch the visualization of these terms in this video by Epic IQ.

Thus, without writing a word, we can prove that, the sum of this infinite series is, 1.

Math is Fun

Can you prove the same result visually, using a linear scale?

Can you also prove it using a rectangle?

Can you also prove it by repeatedly dividing a circle with area one?

More Proof Without Words

A similar geometric series is,

1/4 + 1/16 + 1/64 + …

sum of infinite geometric series, 1/4 to the power n, where n goes from 1 to infinity
sum of infinite series

What will be the sum of this infinite series? There are more than one ways of proving it geometrically. Can you think of it?

Watch the Visual Proof

Sum of First n Odd Integers

Prove it visually, that the sum fist n odd integers, will always be a square!

The theorem states that, the sum of the first n odd integers is always a perfect square and is equal to n^2.  Check the visual proof here.

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