A Famous Infinite Series
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 - whose value 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 - 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 as a Taylor series. The first three 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. The next section, Euler's Infinite Product, explains how Euler came up with this expression.
Next, Euler expanded this product as a new infinite series, whose first two terms are shown below.
Note that the coefficient of x2 is itself an infinite series - in fact, it is the series whose sum Euler was trying to find, multiplied by -1/π2.
Euler 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 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, he knew that a finite polynomial, whose roots are r1, r2, ..., rn and whose constant term is 1, factors as a product of the form
Euler then made a bold leap of faith. He regarded the series for sin(x)/x as an infinite "polynomial," and 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, ±π, ± 2π, ± 3π, 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 is
So Euler rewrote the infinite product as
Next, Euler expanded this infinite product into a series. The constant term in the series is the product of the constants in each expression in parentheses. Since these are all 1, the first term of the series is 1.
But it was the next term in the series - the x2 term - that Euler was really interested in. You get this term by multiplying the x2 term in each expression in parentheses by all the constant terms 1 in every other expression, and adding the results. So the x2 term in the series is the sum of all the x2 terms in the infinite product.
After factoring out -x2 from each of the x2 terms, you get the first two terms of the expanded series:
As described earlier, Euler equated the coefficient of the x2 term in this series with the coefficient of the x2 term in the Taylor series for sin(x)/x, which is -1/6.
Finally, Euler multiplied both sides of this equation by -π2 to obtain his remarkable result.
Copyright 2007 by Paul Trow