Dropping 9-terms from the harmonic series

[Rajeev Joshi told me this problem.]

The harmonic series–that is, 1/1 + 1/2 + 1/3 + 1/4 + …–diverges. That is, the sum is not finite. This is in difference to, for example, a geometric series–like (1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^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 1/9, 1/19, 1/90, 1/992, 1/529110. 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 1/99, 1/992, 1/299, but not 1/9 or 1/909. Does this series converge or diverge?