Asymptotic expansion for the Bell numbers

Question

The Bell numbers $B(n)$ (that is, the numbers that count the set partitions of a set, and have exponential generating function $\exp(e^x -1)$ ) admit the asymptotic expansion $$\frac{\log B(n)}{n} = \log n - \log \log n -1 +\sum_{i,j} c(i,j)\left(\frac{\log \log n}{\log n}\right)^i \frac{1}{(\log n)^j}$$ See for example de Bruijn, Asymptotic methods in Analysis,2nd ed, p108, or P. Flajolet, Analytic Combinatorics, http://algo.inria.fr/flajolet/Publications/book.pdf, p.562)

Is it known if this asymptotic series is actually a convergent series, at least for large enough values of $n$? Thanks for any information.

Valerio De Angelis

Answer 1

The answer to this question is yes, the series is absolutely convergent for all large enough values of $n$, as I recently found out. It is discussed earlier on in de Bruijn's book "Asymptotic methods in analysis", in chapter 2.