Dropping 9-Terms From the Harmonic Series


I got this problem from Rustan Leino, who got it from Rajeev Joshi.

I solved it and wrote up my solution.


The harmonic series (i.e., $1/1 + 1/2 + 1/3 + 1/4 + \ldots$) diverges. That is, the sum isn't finite. This differs from, e.g., a geometric series like $(1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^3 + \ldots,$ which coverges, i.e., 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 $1/9, 1/19, 1/90, 1/992,$ and $1/529110.$ Does the resulting series converge or diverge?

Solution     Reveal