As every calculus student knows, finding the sum of an infinite series can be challenging. A famous example is the following series, whose sum eluded several of the best mathematicians of the 17th century. 

The problem of finding an exact expression for the sum of this series was first posed in 1644. Two of the foremost mathematicians of the time - Leibniz, a co-inventor of the calculus, and Jacob Bernoulli, one of the founders of probability theory - tried to solve the problem and failed. It took the greatest mathematician of the 18th century, Leonhard Euler, to finally obtain the answer in 1735.

To find the sum of the series, Euler started with a seemingly unrelated function:

Using standard tools of calculus, he expanded the function sin(x)/x as a Taylor series. The first four terms of the series are

Then Euler did something truly brilliant. By an ingenious method, he expressed sin(x)/x as the following infinite product:

You may well wonder how this product, which looks so different from the Taylor series, could also represent the function sin(x)/x. Let's imagine how Euler might have discovered this infinite product. He probably thought of the Taylor series for sin(x)/x as an infinite "polynomial."  He knew that a finite polynomial whose roots are r₁, r₂, ..., r_n, and whose constant term is 1, can be factored as a product of the form

Euler then made a remarkable leap of faith. He assumed that he could factor the series for sin(x)/x - an infinite "polynomial" - as a product of the above form. Since sin(x)/x has infinitely many roots, the product contains infinitely many terms.

The roots of sin(x)/x are the non-zero roots of its numerator, sin(x) - that is,  ±π, ± 2π, ± 3π, and so on. (Note that 0 is not a root of sin(x)/x because the limit of sin(x)/x, as x approaches 0, is 1.) Using these roots, Euler factored the series for sin(x)/x as the following infinite product:    

The product of each pair of terms corresponding to the roots kπ  and -kπ  is

By replacing each pair of terms with the expression on the right, Euler rewrote the infinite product for sin(x)/x as

Next, Euler expanded this infinite product as a new infinite series. The constant term in the expansion is the product of all the 1's in the binomial terms above, which equals 1. There is no x term, because all higher order terms are the result of multiplying at least one x² term. Furthermore, each x² term comes from multiplying a term of the form

by the 1's in every other binomial. So the x² terms make up the following infinite sum.

Factoring x² from each term, and adding the constant term 1, the terms of the expanded product up to x² are

Euler did not need to expand the product any further - for observe that the coefficient of x² is exactly the series he was trying to sum, multiplied by -1/π². So he now had two series for sin(x)/x - the series above and the Taylor series.

Since both series represent the same function, the coefficients of x² in the two series must be equal. In other words,

Finally, Euler multiplied both sides of this equation by -π² to get the answer he was seeking.

This is a highly unexpected answer. Why should the sum of the reciprocals of the squares of the natural numbers have anything to do with π, the ratio of a the circumference of a circle to its diameter? The beauty of this result is that it reveals hidden and surprising connections between apparently unrelated ideas.