Moments in the exact summation of the curious series of Kempner type

We obtain for the Kempner series (i.e. harmonic series where certain digits are excluded from all denominators, for example the digit 9 in base 10) new representations as geometrically convergent series. The coefficients for these representations involve the power sums on the allowed digits and the moments of an explicitly described measure on the unit interval. These moments can be computed numerically by recurrence. We establish a priori lower and upper bounds for them. This allows converting the theoretical formulas into an efficient numerical algorithm.