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 relatethem 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 proofof the theorem of Allouche, Hu, and Morin [4] 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 generatingfunction formalism and the work of Guibas–Odlyzko on string overlaps.
