First Harmonic Number to exceed 20

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): $272,400,600$
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