Baillie[4] showed that, rounded to 20 decimals, the actual sum is 22.92067661926415034816 (sequence A082838 in the OEIS).
Schmelzer and Baillie[5] found an efficient algorithm for the more general problem of any omitted string of digits.
Kempner's proof of convergence[3] is repeated in some textbooks, for example Hardy and Wright,[6]: 120 and also appears as an exercise in Apostol.
Therefore the whole sum of reciprocals is at most The same argument works for any omitted non-zero digit.
[5] First we observe that we can work with numbers in base 10k and omit all denominators that have the given string as a "digit".
The sequence is not in general decreasing starting with n = 0; for example, for the original Kempner series we have S(9, 0) ≈ 22.921 < 23.026 ≈ 10 ln 10 < S(9, n) for n ≥ 1.
Therefore, with a small amount of computation, the original series (which is the value for j = 1, summed over all k) can be accurately estimated.
Irwin proved that the sum of 1/n where n has at most k occurrences of any digit d is a convergent series.