Mathematics

The Most Beautiful Sum in Mathematics

The best series have unexpected results.

Image courtesy of Monstera by way of Pexels.

In mathematics, there are few things as nice as an infinite sum. If we can, we like to turn them into geometric sums, and if we can’t, well, we try to figure out why.

One famous summation problem is the Basel problem. The Basel problem is a natural question to ask from the perspective of early calculus/analysis, because it comes up so often. It was first posed by Pietro Mengoli in the early 1600s, and remained open for roughly 100 years until Euler produced a solution. Along the way to this solution, we will encounter some of our favorite characters in math.

Note: for this article, you will need to have a working knowledge of differential calculus, and some basic familiarity with complex numbers. I will provide some resources for the complex analysis portion.

The article will have two portions, and should be the first of several on interesting features of series, and the logarithmic derivative.

A whirlwind introduction to summation by geometric series

Let’s start here: say I want to find fraction of the square covered by red in Figure 1.