>[!abstract] >The **Kempner constant** $\approx22.92067661926415034816\ldots$ is obtained from the summation of the harmonic series >$\sum_{n=1}^{\infty}\frac1n$ >but excluding all values of $n$ that contain the digit $9$. What is surprising about this is that just removing values of $n$ that contain the digit $9$ is enough to cause the otherwise divergent harmonic series to converge. Intuitively, values of $n$ that contain the digit $9$ should account for only $10\%$ of all values of $n$, since $9$ accounts for $10\%$ of the digits $0$ through $9$. In practice, this is not the case, as we can see by counting those values at different intervals of length of $n$: - For $n$ with $1$ digit : only the number $9$, so >[!related] >- **North** (upstream): — >- **West** (similar): — >- **East** (different): — >- **South** (downstream): —