# Harmonic Number Calculator

URL: https://speedycalculator.com/calculators/math/harmonic-number/
Last updated: 2026-10-30

> The harmonic number calculator finds Hₙ = 1 + 1/2 + 1/3 + … + 1/n as an exact fraction (for small n) and a decimal, with the ln n + γ approximation.

## Inputs

- n

## Results

- Hₙ (decimal)
- Hₙ as a fraction
- Approximation

## Formula

```
Hₙ = Σₖ₌₁ⁿ 1/k ≈ ln n + γ + 1/(2n),  γ ≈ 0.5772156649
```

## Explanation

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.

## 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 + …

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Harmonic number](https://mathworld.wolfram.com/HarmonicNumber.html)