We know that the series $\sum_{k=1}^{\infty}\frac{1}{k}$ diverges.
The series $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{8}+\frac{1}{10}+\frac{1}{11}+\dots+\frac{1}{18}+\frac{1}{20}+\frac{1}{21}+\dots+\frac{1}{88}+\frac{1}{100}+\frac{1}{101}+\dots$ converges slowly to approximately $22.92$. What is the exact value of the series?
The given series is the same as $\sum_{k=1}^{\infty}\frac{1}{k}$, but excluding all terms which involve the digit $9$.
