basic

Harmonic Series Partial Sum H_n Calculator

Hₙ = 1 + 1/2 + 1/3 + … + 1/n ≈ ln n + γ.

Compute the n-th partial sum of the harmonic series Hₙ = Σₖ₌₁ⁿ 1/k. Also reports the Euler-Mascheroni approximation Hₙ ≈ ln(n) + 0.5772 and the absolute error. The series diverges but grows extremely slowly (H₁₀₀ ≈ 5.19).

Daniel Oh By Daniel Oh
Published Last reviewed 1 min read

Inputs

Results

Enter values and click Calculate to see results.
Was this helpful?

How to use this calculator

  1. Fill in the inputs above using the units you already have.
  2. Values update automatically as you type — no submit button needed.
  3. Hover any result row for the underlying formula and intermediate values.

Formula

Hₙ = Σ 1/k ≈ ln n + 0.5772.

In depth

Compute the n-th partial sum of the harmonic series Hₙ = Σₖ₌₁ⁿ 1/k. Also reports the Euler-Mascheroni approximation Hₙ ≈ ln(n) + 0.5772 and the absolute error. The series diverges but grows extremely slowly (H₁₀₀ ≈ 5.19).