Euler’s Nonstandard Nonsense


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:

\displaystyle 1 \,+\, \frac{1}{4} \,+\, \frac{1}{9} \,+\, \frac{1}{16} \,+\, \frac{1}{25} \,+\, \cdots \;=\; \frac{\pi^2}{6}

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:

6 x^3 + 3x^2 - 24 x - 12 \;=\; 6 (x-2)(x+\frac{1}{2})(x+2)

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”:

\sin x \;=\;  x \,-\, \frac{1}{6} x^3 \,+\, \frac{1}{5!}x^5 \,-\, \cdots

The roots of this function are just 0,\pm\pi,\pm 2\pi,\ldots, so perhaps there is an infinite product for sine of the form:

\sin x \;=\; Ax(x - \pi)(x + \pi)(x - 2\pi)(x + 2\pi)\cdots

The tricky part is finding the constant A. Usually A is the coefficient of the highest-order term, but the power series for sine has no highest-order term. Indeed, since the coefficients of x^n are approaching zero, there is a good argument that A should be infinitesimal.

After a bit of thought, Euler spotted a solution to this problem. We know that:

\displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1

But clearly:

\displaystyle \lim_{x\rightarrow 0} \frac{x(x - \pi)(x + \pi)(x-2\pi)\cdots}{x} = (-\pi)(\pi)(-2\pi)(2\pi)\cdots

so A should be the reciprocal of the infinite product on the right. This leads to the following guess:

\displaystyle\sin x \;=\; x\left( 1 - \frac{x}{\pi} \right) \left( 1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} \right) \left( 1 + \frac{x}{2\pi} \right) \cdots

\displaystyle\;\;\;\;=\; x\,\prod_{n=1}^{\infty}\left( 1 - \frac{x^2}{(n\pi)^2} \right)

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 x=\pi/2:

\displaystyle 1 \,=\, \frac{\pi}{2} \prod_{n=1}^{\infty} \left( 1 - \frac{1}{4n^2} \right)

Solving for \pi gives a nice product formula (the Wallis product):

\displaystyle \pi \;=\; 2 \prod_{n=1}^{\infty} \frac{(2n)(2n)}{(2n-1)(2n+1)}

The Promised Non-Proof
Euler, however, was more interested in the relationship between the product and the Taylor series. Consider the equation:

\displaystyle x\left(1-\frac{x^2}{\pi^2}\right) \left(1-\frac{x^2}{4\pi^2}\right) \left(1-\frac{x^2}{9\pi^2}\right) \cdots = x - \frac{1}{6}x^3 + \frac{1}{5!}x^5 - \cdots

Presumably the infinite product on the right should “multiply out” to give the infinite sum on the left. Equating the coefficients of x^3 gives the following remarkable equation:

\displaystyle-\frac{1}{\pi^2} \,-\, \frac{1}{4\pi^2} \,-\, \frac{1}{9\pi^2} \,-\, \cdots \;=\; -\frac{1}{6}

Multiply through by -\pi^2 to get \displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.

More p-Series
It is possible to get even more information about p-series this way. Equating the coefficients of x^5 gives the formula:

\displaystyle\sum_{m<n} \frac{1}{m^2 n^2} \,=\, \frac{\pi^4}{120}

It follows that:

\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^4} \;=\; \left(\sum_{m=1}^{\infty} \frac{1}{m^2}\right) \left(\sum_{n=1}^{\infty} \frac{1}{n^2}\right) \,-\, 2\left(\sum_{m<n} \frac{1}{m^2 n^2}\right)

=\; \displaystyle\left(\frac{\pi^2}{6}\right)\left(\frac{\pi^2}{6}\right) \,- \,2\left(\frac{\pi^4}{120} \right) \;=\; \frac{\pi^4}{90}

Using a similar method, one can derive correct formulas for \displaystyle\sum \frac{1}{n^k} for every even value of k. Interestingly enough, very little is known about \displaystyle\sum\frac{1}{n^k} for odd values of k. For example, it is conjectured that the sum is irrational for every odd value of k, but this has only been proven for k=3 (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.

About a year ago, I was browsing around on MathSciNet when I stumbled upon this review of a paper entitled “What is Nonstandard Analysis?” It had the most marvelous sentence:

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.

3. At the end of his post, he mentions a non-standard proof of Gromov’s theorem on groups of polynomial growth. Wow.

As soon as I have time, I’m hoping to go through both Luxemburg’s paper and the van den Dries and Wilkie paper, as well as any other cool but nonstandard papers that present themselves.

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.

20 Responses to “Euler’s Nonstandard Nonsense”

  1. Maya Incaand Says:

    Re NSA

    Now if you could just persuade the mathematical (teaching) establishment that it is worthy of some attention……..

    • Insall Matt Says:

      I used Keisler’s calculus text, along with a traditional advanced calculus text, to teach advanced calculus this semester. The students liked the course, and I introduced them to some of the more advanced topics in nonstandard analysis, in part because they asked, and in part because Keisler’s text is aimed at freshmen, and this class was comprised mainly of juniors and seniors.

  2. Isabel Says:

    The book Euler: The Master of us All has a lot of examples of things like this which Euler did which turned out to be what one might call “rigorizable”. See my blurb about that book, which I’ve updated to include a link to this post.

  3. Greg Muller Says:

    …Nonstandard analysis was already number 50 on the list.

  4. Jim Belk Says:

    Really? I thought I had left it off. In any case, it should be higher up.

    I need to do something about that list becoming too long.

  5. Jonathan Vos Post Says:

    Glad to see W. A. Luxemburg linked to. He was my Math advsor at Caltech (1968-1973), and is still active there as Professor Emeritus.

  6. physicist Says:

    I am confused by something: if I look at the x^1 rather than the x^3 term in your expansion for sin x, I find that


    But every reasonable way of regulating this divergent sum I know of (e.g. via analytic continuation of the zeta function) gives -1/2.

    so why does the expansion of sin x give a non-standard regularization of this sum?

  7. Jim Belk Says:

    Check again. The expansion of the Euler-Wallis formula has only one x^1 term, and its coefficient is the product

    1 x 1 x 1 x . . . = 1

  8. physicist Says:

    oops! you are right of course

  9. Anton Says:

    Some of the things you mentioned are not really non-proofs. For example you can really multiply the infinite products above and they converge locally unformely so the taylor expansions do multiply and you indeed get a holomorphic function whose only zeroes are k \pi, etc

  10. Jim Belk Says:

    Yes indeed. The main part of the proof that’s difficult to make rigorous using classical methods is the derivation of the infinite product for sin x. You can prove it using complex analysis, and I also hear that some complicated elementary treatments exist, but I’m curious whether non-standard analysis is capable of sweeping all of the details under the rug.

    (Important philosophical principle: Details belong under rugs.)

    On a worrisome note, I recently discovered the review for this paper, which seems to indicate that Luxemberg’s account may not be the whole story. From the review:

    The question of the correctness of Euler’s proof, from the point of view of nonstandard analysis (which uses infinite constants at the level of strong mathematical rigor), was already considered by W. A. J. Luxemburg. The author states that Luxemburg “avoided in his analysis some of the most subtle moments in Euler’s judgment”. This sentence may be taken as a motivation for the paper.

    I haven’t yet read either paper, so I can’t speak to whether the criticism is justified.

  11. Sum Divergent Series, III « The Everything Seminar Says:

    […] the fact (demonstrated nicely by Jim on this very blog) that . What do we […]

  12. Mark Says:

    Don’t bother with Nonstandard Analysis (NSA); its just Limit theory in disguise. IMO the best version of analysis is Smooth Infinitesimal Analysis (SIA), which uses nilsquare infinitesimals and the concept of microstraightness. The best book on the subject is A Primer of Infinitesimal Analysis by J L Bell. SIA is better than NSA / Limit theory because:
    1. In SIA differential calculus can be reduced to simple algebra.
    2. SIA facilitates the use of microadditivity in physical derivations.
    3. NSA / Limit theory leads to contradictions like the Banach-Tarski paradox
    whereas SIA does not.
    4. SIA does not employ the arbitrary ‘taking the standard part’ trick of NSA.
    The logic of SIA is Intuitionistic or Constructive logic. It does not contain the Axiom of Choice or the general applicability of the law of excluded middle. These are required for Cardinal numbers, so you can’t believe in infinite and infinitesimal numbers at the same time (see The Foundation of Mathematics, Stewart and Tall).

  13. S. Kutateladze Says:

    “Euler’s nonsense” is an oxymoron, and ignorance is not
    an argument:
    Kanovei V.G.
    Correctness of the Euler method of decomposing the sine function into an infinite product. (Russian)
    Kanovei, V. G.
    Uspekhi Mat. Nauk 43 (1988), no. 4(262), 57–81, 255; translation in Russian Math. Surveys 43 (1988), no. 4, 65–94, MathSciNet.

  14. Badal Joshi Says:

    “Wallis presumably being the first to prove it rigorously”

    It’s good that you hedged your bets with the the word ‘presumably’. Wallis died before Euler was even born. What he did to have his name associated with the formula is to state the special case for x =\frac{\pi}{2}.

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  16. Alex Vong Says:

    This is really brilliant! Thanks for showing us that how Euler derive his infinite product representation of sine. Although I know this is not a rigorous proof, the step involving lim x –> 0 sin x / x = 1 really convinced me, because only this product will give the limit 1 and having the roots of 0, +-pi, +-2pi… at the same time. This help a lot since I don’t know Fourier series and complex analysis, one only need to know elementary calculus to understand this!

  17. Daniel Korenblum Says:

    Simple visualization animating the first 8 pairs of factors:

    Red lines show the factors containing positive roots and blue lines show the factors containing negative roots as they are added. Dotted lines represent the factors accumulated from previous iterations. Light gray line represents the base factor f(x) = x.

    Green line shows the approximant, notice the reasonable convergence in the interval [-pi, pi] as more factors are added.

  18. Daniel Korenblum Says:

    Small bug – you wrote “Presumably the infinite product on the right should “multiply out” to give the infinite sum on the left. ” but the equation is actually oriented the other way, with the factorized form on the LHS and the summation form on the RHS.

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