Kempner-like Harmonic Series

Inspired by a question asked on the list mathfun, we revisit Kempner-like series, i.e., harmonic sums ∑′1/n where the integers n in the summation have "restricted" digits. First we give a short proof that limk→∞(∑s2(n)=k1/n)=2 log 2, where s2(n) is the sum of the binary digits of the integer n. Then we give a generalization that addresses the case where s2(n) is replaced with sb(n), the sum of b-ary digits in base b: we prove that limk→∞∑sb(n)=k1/n=(2 log b)/(b−1). Finally we indicate that other generalizations could be studied: the sum of digits in base 2 could be replaced with, e.g., the function a11(n) of—possibly overlapping—11 in the base-2 expansion of n, for which one can obtain limk→∞∑a11(n)=k1/n=4 log 2.