[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?

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