Geometric series are some of the simplest examples of infinite series with finite sums, although not all of them have this property. The sum of a geometric series is finite as long as the absolute value of the ratio is less than 1. As the successive terms approach zero, they become insignificantly small. This allows a sum to be calculated despite the series containing infinite number of terms. In this post, we are going to discuss 2 proof without words of the series,

1/4 + 1/ 16 + 1/64 + …

sum of infinite series 1/4 + 1/16 = 1/64
sum of infinite series

It is a geometric series with first term 1/4, and common ratio 1/4.
So, its sum will be,

Calculating sum using the formula
calculating the sum

Visual Proof – 1

We start with a 1×1 square with area 1. Now, we divide the square into 4 equal parts and color each part differently.

Divide square 1 into 4 equal parts

We take one of these part and divide it further into 4 equal parts. The area of each of these parts will be, 1/4 of 1/4, equal to 1/16.

Divide top 1/4 into 4 equal parts of 1/16
Divide one 1/4 into 4 equal parts

We keep dividing one of the top-right square into 4 equal parts. One-fourth of this square will keep giving us the next term of the infinite series.

You can watch the math visualization for this infinite series, in the following video.

This division gives us a square covered in 3 different colors, each of equal area. Each of these color represent the geometric series, 1/4 to the power n. So, each color occupies area equal to 1/3.

Visual proof – 2

The same geometric strategy also works with the triangles. We take an equilateral triangle with area 1, and divide it into 4 equal parts.

Dividing triangle into 4 equal parts - visual proof

Then we take one of the 1/4 part, and divide it further into 4 equal parts. We keep repeating this division into 4 parts. After a while the triangle is covered in 3 different colors with equal area. Each of the these area represent the geometric series, so the sum is, 1/3.

Math is Fun

Can you use the same strategy with a trapezoid?

Archimedes proved this result using a parabola? Can you prove using a parabola that,

1 + 1/4 +1/16 + 1/64 + … = 4/3

More Visual Proofs of Geometric Series

Another simple geometric series is,

1/2 + 1/4 + 1/8 + …

What will be its sum? Can you prove it visually or geometrically?

Check out the visual proof of, difference of squares.

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