  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.
