Last time we saw hints of how the geometric series was trying to tell us that there was some “truth” behind the equation
So how do we get at the truth behind such an audacious claim? One fantastically powerful way to sum divergent series such as this is through regularization: finding an analytic function of through which we can relate the divergent sum to some limit in the complex plane. In particular, we can find reasonable values for surprisingly unreasonable sums like
We know that the canonical limiting procedure (at least since Cauchy got his hands on analysis) does not give a finite value to the sum . However, as Tom Leinster pointed out in the last post, there are other analytic methods that we could use to assign a value to this sum. Perhaps the simplest is the Abel sum: we take a small number and multiply the term by . If the terms of our sum are not growing too quickly, the resulting power series
will have a nonzero radius of convergence in the complex plane. On the disc where this function converges, agrees with the function . As a result, if we analytically continue to the rest of we find that
Now we can let our regularizing parameter get larger and larger to define the Abel sum
The exact same regularization method lets us sum the geometric series for all values of except for , and the answer is exactly what we would expect.
It is not clear that any of this has left the realm of “cute but unimportant” yet, so let me present the divergent sum which first made me a believer. I learned about this result while reading John Baez’s lecture notes on generating functions. Let
be the generating function for the Catalan numbers. Explicitly, the coefficient of is the number of binary rooted trees with leaves. By saying exactly what a tree is in the language of generating functions, you can easily compute that
(or now that you know the answer, you can verify by direct computation that the power series given above is the Taylor series for around zero). Happily, there is no pole at one so the series is summable with this regularization method! And what sum do we get?
Well, if we were happy thinking we could sum positive integers to get a negative, it doesn’t seem like such a huge leap to sum positive integers and get a complex number…
Now here is the beautiful thing that made my Platonic heart flutter: we now know that
which in turn means that it is possible for an isomorphism to exist between 1-tuples and 7-tuples of trees (though not for any nontrivial number smaller than 7!) And, in fact, such an isomorphism exists! This was apparently discovered by Lawvere and written up in the fantastic paper Seven Trees in One by Andreas Blass. The amazing thing is that nobody would have expected such an isomorphism to exist if it were not for this seemingly phony calculation using divergent series. So not only can we give these series finite values, but these values can lead us towards new and surprising facts about other standard mathematical objects that we know and love. These regularized divergent sums are not off in some far-flung corner of the mathematical universe — they are intimately connected with math as concrete as combinatorics, and give us honestly new insight into these areas.
Before we get too carried away with this regularization scheme, let us look at some of the things that we still can’t sum. The sum regularizes to the function which has a singularity at , so even the regularized sum appears to diverge. On the other hand, we can do the sum
to prove that . Now, let us do a little bit of arithmetic: the reader can verify that
The right side of this equation allows us to plug in and, using the previously-computed , we find that
though it is entirely unclear what universe this spacey calculation happened in.
Another simple sum that troubles us is . Using our regularization method, this gives
which still has an honest singularity at . However, we can do another spacey calculation to compute this value. Let us begin with the regularized sum
Then we can honestly compute
which is a true statement about the meromorphic functions and . But now we can let approach 1 to get
proving(?) the infamous formula
Coming up next: zeta regularization and the promised puzzle! (apologies for dragging it across three posts, but the post length was getting a bit too divergent…) In the meantime, please check out John Baez’s fantastic lectures on generating functions, structure types, and summing divergent series for fun and profit here and the recent discussion on divergent sums and Tom Leinster’s work on generalized Euler characteristics (one of the best motivations for paying attention to divergent sums) at the n-Category Cafe. If you’ve got a lot of time on your hands, there is an entire fascinating book by G. H. Hardy called Divergent Series available at the Internet Archive. The calculation of above comes from the first few pages of chapter 1.
Finally, something that has been puzzling me for quite some time. When I showed Josh Bowman the generating function
he asked me “is there a reason that is so close to the inverse of the Koebe function?” To be precise, and making the Euclidean substitution yields the Koebe function . Now: what on earth do binary trees have to do with conformal mappings?