by Paul Trow
| 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, ..., rn, 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. |
