Summation of Infinite Geometric Series, Part 2

By Chandrahas M. Halai

You might be wondering who must have been the first mathematician to calculate the sum of this infinite geometric series?

It was Virasena who had done so in his commentary Dhavala (816 CE) on the Digambara Jain text Shatkhandagama. He had calculated the sum of this infinite series to evaluate the volume of the frustum of a right circular cone.

This summation was proved by Nilakantha Somayaji (1444-1544 CE) (mathematician from Kerala in south India) in Aryabhatiyabhasya his commentary on Aryabhatiya. For details refer my earlier article:

https://chandrahasblogs.wordpress.com/2021/07/25/summation-of-infinite-geometric-series/

In this article I am going to derive the sum of the given infinite geometric series by geometric means.

Diagram 1

Step 1

Draw an equilateral triangle having an area 1 as shown in Diagram 1.

Diagram 2

Step 2

Join the midpoints of the sides of the equilateral triangle as shown in Diagram 2. This will divide the triangle into four congruent (same size) equilateral triangles. Now the areas of each of these four triangles will be 1/4.

Diagram 3

Step 3

Now join the midpoints of the sides of the top (white) triangle as shown in Diagram 3. This will divide the top triangle into four congruent (same size) equilateral triangles. Now the areas of each of these four new triangles will be 1/4^th of ¼.

Now, the sum of the areas of each of the coloured triangles (blue, black and green) will be

Diagram 4

Step 4

Now join the midpoints of the sides of the top (white) triangle as shown in Diagram 4. This will divide the top triangle into four congruent (same size) equilateral triangles. Now the areas of each of these four new triangles will be 1/4^th of .

Now, the sum of the areas of each of the coloured triangles (blue, black and green) will be

Diagram 5

Step 5

Now join the midpoints of the sides of the top (white) triangle as shown in Diagram 5. This will divide the top triangle into four congruent (same size) equilateral triangles. Now the areas of each of these four new triangles will be 1/4^th of .

Now, the sum of the areas of each of the coloured triangles (blue, black and green) will be

Diagram 6

Step 6

Now join the midpoints of the sides of the top (white) triangle as shown in Diagram 6. This will divide the top triangle into four congruent (same size) equilateral triangles. Now the areas of each of these four new triangles will be 1/4^th of .

Now, the sum of the areas of each of the coloured triangles (blue, black and green) will be

Step 7

Repeat the above procedure ad infinitum.

Then, the sum of the areas of each of the coloured triangles (blue, black and green) will be

We can observe from diagram 6 that more the terms you add to the series the closer it gets to 1/3. Hence, we can say that the given series converges to 1/3.