[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?