Mathematics July 22, 2026 · 10 min read

Harmonic Numbers: The Slowly Diverging Series

Understand harmonic numbers H(n) = 1 + 1/2 + 1/3 + ... + 1/n. Learn the formulas, approximations, and connections to the Euler-Mascheroni constant.

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The harmonic numbers — defined as H(n) = 1 + 1/2 + 1/3 + ... + 1/n — are a sequence of numbers that grows without bound, but does so incredibly slowly. They connect to some of the deepest results in mathematics, from the divergence of the harmonic series to the Riemann zeta function.

Definition

H(n) = Σ 1/k from k=1 to n

Asymptotic Approximation

H(n) ≈ ln(n) + γ + 1/(2n) - 1/(12n²)

Where γ ≈ 0.5772 is the Euler-Mascheroni constant. This approximation is remarkably accurate even for moderate values of n.

The Divergence of the Harmonic Series

Despite the terms approaching zero, the harmonic series diverges — its sum grows without bound. This was proven by Nicole Oresme around 1350 using a clever grouping argument.

Worked Example

Calculate H(5):

  1. H(5) = 1 + 1/2 + 1/3 + 1/4 + 1/5
  2. = 1 + 0.5 + 0.333 + 0.25 + 0.2
  3. = 2.2833...

Applications

  • Algorithm analysis: Expected comparisons in quicksort
  • Probability: Coupon collector problem
  • Number theory: Connections to the Riemann zeta function