Kempner series

The Kempner series is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit '9'. That is, it is the sum

where the prime indicates that n takes only values whose decimal expansion has no 9s. The series was first studied by A. J. Kempner in 1914. The series is interesting because of the counter-intuitive result that, unlike the harmonic series, the Kempner series converges. Kempner showed the sum of this series is less than 80. Baillie showed that, rounded to 20 decimals, the actual sum is 22.92067 66192 64150 34816 (sequence A082838 in OEIS)).