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We provide an exact geometrically convergent formula for the summation of the general Kempner series, i.e. the sum of all reciprocals allowed only certain digits in a given base. The coefficients are built from finite power sums and moments of a certain measure. The moments obey a linear recurrence and a priori bounds which means the theory can be immediately converted into an efficient numerical algorithm.
Jean-François Burnol (2024) studied this question.