First Harmonic Number to exceed 20
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Theorem
The first harmonic number that is greater than $20$ is $H_{272 \, 400 \, 600}$.
That is, the number of terms of the harmonic series required for its partial sum to exceed $20$ is $272 \, 400 \, 600$.
Proof
We have:
- $H_{272 \, 400 \, 599} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 599} \frac 1 k \approx 19 \cdotp 99999 \, 99979$
and:
- $H_{272 \, 400 \, 600} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 600} \frac 1 k \approx 20 \cdotp 00000 \, 00016$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $272,400,600$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $272,400,600$