Portrait or Leonhard Euler by Jakob Emanuel Handmann (1753) Source

Euler’s Solution to the Basel Problem

The proof is short and accessible. Yet it evaded mathematicians for nearly a century.

The Basel Problem is simple to state. Yet it stumped mathematical minds for 90 years. A 28-year-old physics professor named Leonhard Euler turned heads when he published a solution in 1734. Here is the problem and Euler’s solution.

The problem

What is the sum of the reciprocals of the squares of the positive integers?

We can plot the first 10 partial sums. That seems like a steady rise in values toward the end (Figure 1). Does this series continue to climb forever?

Figure 1: The first 10 partial sums of the Basel Problem. How many more sums would you plot before you felt certain that the sum never crosses the dotted red line?

Setting forever aside, where do you think we would end up after an additional 10 partial sums? Would we cross the dotted red line? How many sums would you take before you decided that the series never crosses the line?