Harmonic numbers grow without bound but very slowly – you need over 12,000 terms to pass 10. They appear in the coupon collector problem, the analysis of quicksort, the stacking-blocks overhang puzzle and the “jeep problem”.
How to use the Harmonic Number Calculator
- n
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
For small n the terms are added exactly as fractions; for large n the decimal sum and the asymptotic formula are given.
Formula
Hₙ = Σₖ₌₁ⁿ 1/k ≈ ln n + γ + 1/(2n), γ ≈ 0.5772156649
Worked example
Inputs — n: 10.
Results — Hₙ (decimal) = 2.92896825397; Hₙ as a fraction = 7381/2520; Approximation = ln n + γ + 1/(2n) ≈ 2.929800758.
Key terms
- Harmonic series
- 1 + 1/2 + 1/3 + …
Frequently asked questions
Does the harmonic series converge?
No – it diverges, though extremely slowly.
What is H₁₀?
7381/2520 ≈ 2.929.
What is the coupon collector problem?
Collecting all n coupons takes about n·Hₙ draws on average.
What is γ?
The Euler–Mascheroni constant, ≈ 0.5772.