The Basel problem asks for the precise summation of the reciprocals of the squares of positive integers, i.e. the precise sum of the infinite series:
It asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be and announced this discovery in 1735.
Dividing by , we have
Using the Weierstrass factorization theorem, it can also be shown that the left-hand side is the product of linear factors given by its roots, just as we do for finite polynomials:
If we formally multiply out this product and collect all the terms (we are allowed to do so because of Newton's identities), we see that the coefficient of is
But from the original infinite series expansion of , the coefficient of is . These two coefficients must be equal, so
Multiplying through both sides of this equation by gives the sum of the reciprocals of the positive square integers: