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 - the sum of the reciprocals of the positive integers squared.
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 - 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 this function as a Taylor series. The Taylor series for sin(x) is
Dividing each term by x gives
Then Euler did something truly brilliant. By an ingenious method, he expressed the same function, sin(x) / x, as the following infinite product of binomials:
You may well wonder how this product, which looks so different from the Taylor series, could also represent the function sin(x) / x. In the next section, we'll take a look at how Euler came up with this infinite product. But for the moment, just take it on faith that the product equals sin(x) / x.
Euler's next step was to expand the infinite product for sin(x) / x by multiplying its terms, just as you would expand a finite product of binomials. The first term of this expansion is 1, which is the result of multiplying all the 1's in the binomials together.
The next terms in the expansion are the really important ones - the x2 terms. Each of these is the result of taking one of the terms of the form
, , , , ...from each binomial and multiplying it by the constant term 1 in all the other binomials. Adding all of these terms together gives
After factoring out from each term, the x2 term of the expanded product is
Adding this to the constant term 1 gives the first two terms of the series expansion:
Note that the coefficient of x2 is itself an infinite series - in fact, it is precisely the series Euler was trying to sum, multiplied by -1 / π2.
Euler now had two series for sin(x)/x - the series above and the original 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 beautiful and surprising result - there is no reason to expect π, the ratio of the circumference of a circle to its diameter, to appear in the result, since the sum of the reciprocals of the integers squared has no apparent connection with geometry.
Let's take a closer look at how Euler found the infinite product for sin(x) / x. First, by elementary algebra, 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 bold leap of faith. Looking at the series for sin(x) / x as an infinite "polynomial," he assumed he could factor it 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, and so on.
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, where k is a positive integer, is
Multiplying out each pair of terms for all positive integers k, Euler rewrote this expression as
which is the infinite product for sin(x) / x described in the previous section.
Copyright 2007 by Paul Trow