Harmonic Number Calculator H(n)

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Standard computes H(n) = 1 + 1/2 + ... + 1/n. Generalized computes H(n,m) = 1 + 1/2^m + ... + 1/n^m.
Positive number for standard mode (integers give an exact fraction). Positive integer only for generalized mode.
H(n)H(n) 2-3 (moderate n)
2.92896825

The nth harmonic number (exact decimal)

Exact fraction7381/2520
ln(n) + gamma approximation2.92896742
Approximation error-0%
H(n) exact2.92896825
ln(n) + gamma approx2.92896742

H(10) = 2.928968 - the sum of the first 10 unit fractions.

  • H(10) = 2.928968, which is approximately ln(10) + 0.5772 = 2.879801.
  • Doubling n to 20 only adds about ln(2) = 0.693 to the sum, giving H(20) = 3.5977. This slow logarithmic growth is why the series diverges but extremely slowly.
  • The asymptotic approximation ln(n) + gamma + 1/(2n) - 1/(12n^2) is within 0.0000% of the exact value, becoming more accurate as n grows.

Next stepThe harmonic series diverges to infinity, but you need H(n) > 10 only at n around 12367, and H(n) > 20 only at around n = 272 million.

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