First Harmonic Number to exceed 10

Theorem

The first harmonic number that is greater than $10$ is $H_{12 \, 367}$.

That is, the number of terms of the harmonic series required for its partial sum to exceed $10$ is $12 \, 367$.

Proof

We have:

$H_{12 \, 366} = \ds \sum_{k \mathop = 1}^{12 \, 366} \frac 1 k \approx 9 \cdotp 99996 \, 214$

and:

$H_{12 \, 367} = \ds \sum_{k \mathop = 1}^{12 \, 367} \frac 1 k \approx 10 \cdotp 00004 \, 30083$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $272,400,600$