Ellipsephic harmonic series revisited

Ellipsephic or Kempner-like harmonic series are series of inverses of integers whose expansion in base $B$, for some $B \geq 2$, contains no occurrence of some fixed digit or some fixed block of digits. A prototypical example was proposed by Kempner in 1914, namely the sum inverses of integers whose expansion in base $10$ contains no occurrence of a nonzero given digit. Results about such series address their convergence as well as closed expressions for their sums (or approximations thereof). Another direction of research is the study of sums of inverses of integers that contain only a given finite number, say $k$, of some digit or some block of digits, and the limits of such sums when $k$ goes to infinity. Generalizing partial results in the literature, we give a complete result for any digit or block of digits in any base.