First Harmonic Number to exceed 100

Theorem

The first harmonic number that is greater than $100$ is $H_n$ where $n \approx 1.5 \times 10^{43}$.

That is, it takes approximately $1.5 \times 10^{43}$ terms of the harmonic series required for its partial sum to exceed $100$.


Proof


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Sources

  • Oct. 1971: R.P. Boas, Jr. and J.W. Wrench, Jr.: Partial Sums of the Harmonic Series (Amer. Math. Monthly Vol. 78, no. 8: pp. 864 – 870)  www.jstor.org/stable/2316476
  • 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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