Matt’s post on the harmonic series has inspired me to share with you some nonsense due to Euler, leading to a beautiful non-proof of the following identity:
In my opinion, this is one of the few non-proofs that every mathematician should know.
An Infinite Product
Euler discovered the identity while thinking about polynomials. Recall that any polynomial can be expressed as either a sum or as a product:
Given the form on the left, you need only find the roots of the polynomial to obtain the form on the right.
Euler wondered whether this might work with the following “polynomial”:
The roots of this function are just , so perhaps there is an infinite product for sine of the form:
The tricky part is finding the constant . Usually is the coefficient of the highest-order term, but the power series for sine has no highest-order term. Indeed, since the coefficients of are approaching zero, there is a good argument that should be infinitesimal.
After a bit of thought, Euler spotted a solution to this problem. We know that:
so should be the reciprocal of the infinite product on the right. This leads to the following guess:
The amazing thing is that this product formula actually works! It’s called the Euler-Wallis formula for sine, Wallis presumably being the first to prove it rigorously. Today you can find it in complex analysis books as a corollary to the Weierstrass factorization theorem.
A Product for Pi
The above formula has a whole range of applications. For example, watch what happens if you plug in :
Solving for gives a nice product formula (the Wallis product):
The Promised Non-Proof
Euler, however, was more interested in the relationship between the product and the Taylor series. Consider the equation:
Presumably the infinite product on the right should “multiply out” to give the infinite sum on the left. Equating the coefficients of gives the following remarkable equation:
Multiply through by to get .
It is possible to get even more information about -series this way. Equating the coefficients of gives the formula:
It follows that:
Using a similar method, one can derive correct formulas for for every even value of . Interestingly enough, very little is known about for odd values of . For example, it is conjectured that the sum is irrational for every odd value of , but this has only been proven for (see the Wikipedia articles on zeta constants and Apéry’s constant).
Is This Really Nonsense?
First of all, I should state for the record that all of the above formulas have been proven rigorously. (See this article for a fairly elementary approach.)
That’s not what I’m interested in.
As a particularly interesting example [the author] discusses Euler’s original proof for the product formula for the sinefunction in which Euler, as customary for the founders of the calculus, freely used expressions such as “infinitely large” and “infinitely close”. The author shows how Euler’s proof in a natural way can be made precise within the context of nonstandard analysis.
I briefly looked at the paper, but the description of nonstandard analysis seemed somewhat technical, so I filed it and went back to learning geometry.
Recently, however, I read and understood Terence Tao’s wonderful post on nonstandard analysis using ultrafilters. This has made me very excited about learning nonstandard analysis, for the following reasons:
1. It is supremely cool. (This has always been true. See above.)
2. It is related to ultrafilters and ultralimits. These things are good friends of mine, since they are useful for understanding the amenability of groups.
Of course, there are already an enormous number of subjects that I’d like to learn. I guess I’ll just have to add this to the list.