  We consider the harmonic series $S(k)=\sum^{(k)} m^{-1}$ over the integers
having $k$ occurrences of a given block of $b$-ary digits, of length $p$, and
relate them to certain measures on the interval $[0,1)$. We show that these
measures converge weakly to $b^p$ times the Lebesgue measure, a fact which
allows a new proof of the theorem of Allouche, Hu, and Morin which says $\lim
S(k)=b^p\log(b)$. A quantitative error estimate will be given. Combinatorial
aspects involve generating series which fall under the scope of the
Goulden-Jackson cluster generating function formalism and the work of
Guibas-Odlyzko on string overlaps.
