Dropping 9-terms from the harmonic series

[Rajeev Joshi told me this problem.]

The harmonic series–that is, 11 + 12 + 13 + 14 + …–diverges. That is, the sum is not finite. This is in difference to, for example, a geometric series–like (12)^0 + (12)^1 + (12)^2 + (12)^3 + …–which converges, that is, has a finite sum.

Consider the harmonic series, but drop all terms whose denominator represented in decimal contains a 9. For example, you’d drop terms like 19, 119, 190, 1992, 1529110. Does the resulting series converge or diverge?

[More generally, you may consider representing the denominator in the base of your choice and dropping terms that contain a certain digit of your choice.]

[Here is a follow-up question suggested by Gary Leavens.]

Consider again the harmonic series, but drop a term only if the denominator represented in decimal contains two consecutive 9’s. For example, you’d drop 199, 1992, 1299, but not 19 or 1909. Does this series converge or diverge?

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