The sum of the series is approximately equal to 1.644934A013661. The Basel problem 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 π2/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he was later proven correct, and it was not until 1741 that he was able to produce a truly rigorous proof.
Euler's original derivation of the value π2/6 essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series.
Of course, Euler's original reasoning requires justification (100 years later, Weierstrass proved that Euler's representation of the sine function as an infinite product is correct, see: Weierstrass factorization theorem), but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series. The agreement he observed gave him sufficient confidence to announce his result to the mathematical community.
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 (which Euler assumed, but is not always true):
If we formally multiply out this product and collect all the x2 terms (we are allowed to do so because of Newton's identities), we see that the x2 coefficient of sin(x)/x is
But from the original infinite series expansion of sin(x)/x, the coefficient of x2 is −1/(3!) = −1/6. These two coefficients must be equal; thus,
Multiplying through both sides of this equation by -π2 gives the sum of the reciprocals of the positive square integers.
The Riemann zeta function ζ(s) is one of the most important functions in mathematics, because of its relationship to the distribution of the prime numbers. The function is defined for any complex numbers with real part > 1 by the following formula:
Taking s = 2, we see that ζ(2) is equal to the sum of the reciprocals of the squares of the positive integers:
Convergence can be proven with the following inequality:
This gives us the upper bound 2, and because the infinite sum has only positive terms, it must converge. It can be shown that ζ(s) has a nice expression in terms of the Bernoulli numbers whenever s is a positive even integer. With s=2n:
The proof goes back to Augustin Louis Cauchy (Cours d'Analyse, 1821, Note VIII). In 1954, this proof appeared in the book of Akiva and Isaak Yaglom "Nonelementary Problems in an Elementary Exposition". Later, in 1982, it appeared in the journal Eureka, attributed to John Scholes, but Scholes claims he learned the proof from Peter Swinnerton-Dyer, and in any case he maintains the proof was "common knowledge at Cambridge in the late 1960s".
The inequality is shown. Taking reciprocals and squaring gives .
The main idea behind the proof is to bound the partial sums
between two expressions, each of which will tend to π2/6 as m approaches infinity. The two expressions are derived from identities involving the cotangent and cosecant functions. These identities are in turn derived from de Moivre's formula, and we now turn to establishing these identities.
Let be a real number with , and let n be a positive odd integer. Then from de Moivre's formula and the definition of the cotangent function, we have
Combining the two equations and equating imaginary parts gives the identity
We take this identity, fix a positive integer , set and consider for . Then is a multiple of and therefore . So,
for every . The values are distinct numbers in the interval (0, π/2). Since the function is one-to-one on this interval, the numbers are distinct for r = 1, 2, …, m. By the above equation, these m numbers are the roots of the mth degree polynomial
By Vieta's formulas we can calculate the sum of the roots directly by examining the first two coefficients of the polynomial, and this comparison shows that
Substituting the identity , we have
Now consider the inequality . If we add up all these inequalities for each of the numbers , and if we use the two identities above, we get
Multiplying through by (π/(2m + 1))2, this becomes
As m approaches infinity, the left and right hand expressions each approach , so by the squeeze theorem,