How to Get PI — Euler Used 2 Beautiful Series

preparing for the π day!

Euler, as much a great mathematician as he was, could obtain two beautiful formulas involving π:

The legendary solution to the Basel problem:

∑(n=1 to ∞) 1/n² = π²/6

A pretty infinite product formula for π:

2/π = ∏(n=1 to ∞)(1 - 1/(4n²))

These 2 results show the profound power that lies in infinite series and infinite products.

Let’s dive into Euler’s masterwork step by step.

The Basel Problem

∑(n=1 to ∞) 1/n²

Here’s what Euler’s thought process was like

Step 1: Taylor Series Expansion for sin(x)

Euler began with the familiar Taylor series expansion:

sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ...

This series is well-known, and each coefficient is precisely determined.

Step 2: Euler’s Insight: Factorization of sin(x)

Euler realized that polynomial functions factor neatly into linear terms, such as:

x² - 4 = (x - 2)(x + 2)

He boldly extended this idea, hypothesizing that transcendental functions like sin(x) could also be expressed as infinite products.

Since sin(x) is zero at precisely x = 0, ±π, ±2π, ±3π, …, Euler hypothesized: