A few days ago I wrote about computing ζ(3). I spent most of that post discussing simple but inefficient methods of computing ζ(3), then mentioned that there were more efficient methods.

I recently ran across a paper [1] that not only gives more efficient methods for computing ζ(3), it gives a method for generating methods. For every positive integer *s*, the “meta method” yields a method of computing ζ(3). As *s* increases, the methods get more complicated but also more efficient.

The simplest method, corresponding to *s* = 1 is

There are several things to say about this series. First, it goes back to Apéry and is not new in [1], but [1] puts it into the context of a sequence of series. Second, the series converges quickly because the binomial coefficient in the denominator grows exponentially, something like 4* ^{n}*. Third, the series alternates, which suggests it could be accelerated [2], though this may not be that important because you could derive series that converge even faster by increasing

*s*.

The series for *s* = 2 is more complicated, but converges faster. It has in its denominator the binomial coefficient (2*n*, *n*) as before, but also the binomial coefficient (3*n*, *n*) which grows even faster. The series for *s* = 3 has the binomial coefficients (4*n*, *n*) and (3*n*, *n*) in the denominator.

How fast do these binomial coefficients grow? Let’s just say “very fast” for now. The next post will get more specific.

## Related posts

[1] Faster and Faster Convergent Series for ζ(3). Available here.

[2] Series acceleration methods, like Aitken acceleration and newer methods, generally work better on alternating series.