Kempner series

Example

The Kempner series is the sum of $1/n$ over the positive integers $n$ whose decimal expansion has no digit $9$: $$\sum_{\substack{n \ge 1 \\ \text{no digit } 9}} \frac{1}{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{8} + \frac{1}{10} + \cdots + \frac{1}{18} + \frac{1}{20} + \cdots.$$ It is a convergent series, and its sum is less than $80$.

Check: a positive integer with exactly $k$ digits and no digit $9$ has a first digit in $\set{1, \ldots, 8}$ and $k - 1$ further digits in $\set{0, \ldots, 8}$. So there are $8 \cdot 9^{k-1}$ such integers, and each is at least $10^{k-1}$. Their reciprocals add up to at most $8 \cdot 9^{k-1} / 10^{k-1} = 8 \cdot (9/10)^{k-1}$. Summing over $k \ge 1$ gives the bound $8 / (1 - 9/10) = 80$. The terms are positive, so the partial sums are increasing and bounded, and the series converges.