Wallis's Product Formula for Pi

When Newton said, “If I have seen a little farther than others it is because I have stood on the shoulders of giants,” one of those giants was John Wallis. Wallis taught at Cambridge before becoming Savilian Professor of Geometry at Oxford. His Arithmetica Infinitorum, published in 1655, derives the rule (found also by Fermat) for the integral of a fractional power of x:

Our interest lies in one of the applications he makes of this rule, an infinite process that yields a value for . It does not involve an infinite series but rather an infinite product. It is included because it demonstrates an audacious use of the infinite, it motivated Newton's calculation of , and it yields a useful formula that will play an important role in estimating the size of n!.

We begin with the observation that is the area of any circle with radius 1. If we locate the center of our circle at the origin, then the quarter circle in the first quadrant

has area

This looks very much like the kind of integral Wallis had been studying. Any means of calculating this integral will yield a means of calculating . Realizing that he could not attack it head on, Wallis looked for similar integrals that he could handle. His genius is revealed in his decision to look at When q is a small positive integer, we can expand the integrand:

A pattern is emerging, and it requires little insight to guess that

where 4! = 4·3·2·1 = 24, 5! = 5·4·3·2·1 = 120, and so on.

These numbers are fractions. So that we can work with integers, we shall follow Wallis and look at the reciprocal of this integral:

Wallis began to construct a table of values for f(p,q), allowing P to stand for

f(p,q)

Our first observation is that whenever we know both f(p,q) and f(q,p) (when they are both integers), then they are equal. This suggests that maybe they are always equal. Wallis made this assumption:

f(p,q)

After staring at this table long enough, one begins to detect a pattern in the ratio of f(p,q) to f(p,q–1):

f(p,q)/f(p,q–1)

Something is going on here. Wallis was able to guess that

This enabled him to finish filling in his table.

f(p,q)

We see that any row in which p is a positive integer is increasing from left to right, and it is reasonable to expect that the row p = 1/2 is also increasing from left to right. Recalling that this implies a string of inequalities:

These, in turn, yield inequalities for :

It is easier to see what is happening with these inequalities if we look at our string of inequalities in terms of the ratios that led us to find them in the first place:

In general, we have that

This yields a general inequality for that we can make as precise as we want by taking n sufficiently large:

As n gets larger, these bounds on approach each other. Their ratio is 2n/(2n+1) which approaches 1. Wallis therefore concluded that

Note that this product alternately grows and shrinks as we take more terms.