Question

It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series, and it's also sort of folklore (although I can't remember where I heard this) that the harmonic series is more or less the only important series with this property.

What other methods besides analytic continuation and zeta regularization exist for summing divergent series? Do they work on the harmonic series? And are there other well-known series which also don't have obvious regularizations?

Answer 1

One common regularization method that wasn't mentioned in the Everything Seminar post is to take the constant term of a meromorphic continuation. While the Riemann zeta function has a simple pole at 1, the constant term of the Laurent series expansion is the Euler-Mascheroni constant gamma = 0.5772156649...

It is reasonable to claim that most divergent series don't have interesting or natural regularizations, but you could also reasonably claim that most divergent series aren't interesting. Any function with extremely rapid growth (e.g., the Busy Beaver function) is unlikely to have a sum that is regularizable in a natural way.

Answer 2

Incidentally, the best text on such questions is Hardy's last book, Divergent Series.

Answer 3

I have a sneaking suspicion that anything that works on 1/2 + 1/3 + 1/5 + ... will probably also work on the harmonic series, although I certainly don't have any hard reasoning to back this up -- just that it doesn't have nice local properties or nice global properties, much like the harmonic series.

But I sort of hope I'm wrong -- I'd be very interested to see what a regularization for this series looks like!

Answer 4

The series 1/2 + 1/3 + 1/5 + ... (the sum of reciprocals of the primes) mentioned by harrison "sums to log log ∞"; more formally,

(1/2 + 1/3 + 1/5 + 1/7 + ... + 1/n) ~ log log n

where ~ has the usual meaning: f(n)~g(n) if lim (n -> infty) f(n)/g(n) = 1.

The nth partial sum of the harmonic series, 1 + 1/2 + 1/3 + ... + 1/n, diverges like log n.

Perhaps sums which diverge "logarithmically fast" are in general problematic, and the harmonic series is just the canonical example of such a series.

Answer 5

I'm not allowed to post a comment, but in reply to Michael Lugo's post and as a followup to Scott Carnahan, the prime harmonic series can be regularized in analogy with $1 + 1/2 + 1/3 + 1/4 + \ldots$ "$=$" $\gamma$, giving the Mertens constant. See the prime zeta function for more information.

In this case it's not "meromorphic continuation" as the singularity is logarithmic. This leads to the followup question: is there a practical difference, and is there a general theory for the logarithmic (or even more general, e.g. multiply nested logarithmic) case? The prime zeta function has some interesting properties, such as having a natural boundary of analyticity at $\Re(s) = 0$.

Answer 6

There are other sums with no good summation: for example 1+1+1+... Any decent method of summation would yield S=1+S.