1 The Basel Problem

In 1735, Euler stunned the mathematical world by computing the exact value of the sum of reciprocal squares, a problem that had resisted attack for nearly a century.

Consider the infinite series:

S=∑n=1∞1n2=1+14+19+116+⋯S = \sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots

Recall the Taylor series for sine and its Weierstrass product:

sin⁡xx=1−x23!+x45!−⋯\frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \cdots
sin⁡xx=∏n=1∞(1−x2n2π2)\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1 - \frac{x^2}{n^2\pi^2}\right)

Expand the product and compare coefficients of x squared:

−13!=−1π2∑n=1∞1n2-\frac{1}{3!} = -\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}
∴
  ∑n=1∞1n2=π26  \boxed{\;\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}\;}

The appearance of pi in a sum involving only integers remains one of the most striking surprises in analysis.