KnowraBasel problemBasel problemThe Basel problem asks for the exact value of the infinite series 1 + 1/4 + 1/9 + ⋯. Its sum is π²/6.BriefConnectEuler's solution to the Basel problem: Leonhard Euler's 1735 evaluation of the sum of reciprocal squares as π²/6. Euler used the infinite product for sine to extract the sum directly.Pietro Mengoli: An Italian mathematician who posed the reciprocal-square summation problem in 1644. His published question began the problem's recorded history.Infinite series: A sum with infinitely many terms, defined through the limit of its finite partial sums. The Basel problem is a question about the limit of partial sums.Zeta function at 2: The value of the Riemann zeta function at 2, equal to π²/6. This is the modern notation for the exact answer to the problem.Fourier series: Representations of periodic functions as sums of sines and cosines. The Fourier series of x² gives a direct route to the sum.Jacob Bernoulli: A Swiss mathematician who studied the reciprocal-square series and challenged others to evaluate it. Bernoulli publicized the unsolved problem and named it for Basel.Convergence of a series: The property that a series' sequence of partial sums approaches a finite limit. The reciprocal-square terms converge, so an exact finite sum exists.Bernoulli numbers: A sequence of rational numbers that appears in power sums, series expansions, and zeta values. They encode the formula that extends Euler's result to higher even powers.Euler's infinite product for the sine function: An infinite product expressing sin x in terms of its zeros at integer multiples of π. Comparing its coefficients with the sine series reveals the reciprocal-square sum.Johann Bernoulli: A Swiss mathematician and Jacob Bernoulli's younger brother, active in analysis and calculus. He was part of the family mathematical circle that pursued the sum.Show all 21Linked from 4 pagesEuler productRelated: Euler’s solution used the zeta function whose prime product became a landmark result.Divergence of the sum of the reciprocals of the primesRelated: Euler’s solution used the zeta function and prime products that also illuminate reciprocal primes.Show all 4