---
tag: 0DMY75D6FV
kind: statement
role: theorem
name: Basel problem
aliases: [sum-of-reciprocals-of-squares]
msc: [40A05]
uses: [0ZNSSK8ZK1, 0C1VYVNV2N]
sources:
  - url: https://en.wikipedia.org/wiki/Basel_problem
    locator: lead section
same_as:
  wikidata: Q810431
  wikipedia: Basel problem
---

## Statement

The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, and
$$\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \cdots = \frac{\pi^2}{6}.$$

## Remarks

The convergence follows from the [[p-series test|0ZNSSK8ZK1]] with $p = 2$. The value is the hard part.
The value $\pi^2/6 \approx 1.6449$ is $\zeta(2)$, where $\zeta$ is the Riemann zeta function.
Mathlib states it as `hasSum_zeta_two`.

## History

Pietro Mengoli posed the problem in 1650.
Leonhard Euler solved it in 1734. The problem is named after Basel, the home town of Euler.

## Explanations

### L1 human

The sum of the reciprocals of the squares $1, 4, 9, 16, \dots$ is a finite number.
Surprisingly, this number involves $\pi$, which comes from circles.