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