Harmonic Number Calculator

The nth harmonic number H(n) is the sum of reciprocals: 1 + 1/2 + 1/3 + ... + 1/n.

No, the harmonic series diverges. However, it diverges very slowly — H(n) grows like ln(n).

γ ≈ 0.5772 is defined as the limit of H(n) - ln(n) as n — ∞. It appears throughout number theory and analysis.

Use H(n) ≈ ln(n) + γ + 1/(2n) - 1/(12n²). This is accurate to many decimal places for large n.

H(1) = 1. It is the first and simplest harmonic number.

H(n) grows logarithmically: H(n) ≈ ln(n) + γ. Doubling n increases H(n) by approximately ln(2) ≈ 0.693.

1 - 1/2 + 1/3 - 1/4 + ... = ln(2). Unlike the regular harmonic series, this converges.

They appear in the analysis of algorithms like quicksort, hash tables, and the coupon collector problem.

The digamma function ψ(n) = H(n-1) - γ is the logarithmic derivative of the gamma function, generalizing harmonic numbers.

No. For positive integers n, H(n) is always positive and increasing.

H(n) = ∫₀¹ (1 - xⁿ)/(1 - x) dx, expressing the harmonic number as an integral.

H(n) is a partial sum of the zeta function ζ(1), which diverges. The zeta function ζ(s) converges for s > 1.

H(p-1) ≡ 0 (mod p) for prime p > 3 (Wolstenholme's theorem).

Yes, using the digamma function: H(x) = ψ(x+1) + γ, which extends harmonic numbers to real and complex values.

For practical purposes, you can compute H(n) for n up to about 10^15 before precision becomes an issue.

The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. The name "harmonic" comes from the harmonic series.

H(n,m) = 1 + 1/2ᵐ + 1/3ᵐ + ... + 1/nᵐ generalizes to power m, connecting to the Riemann zeta function.

H(n) = ln(n) + γ + 1/(2n) - 1/(12n²) + 1/(120n⁴) - ... with terms involving Bernoulli numbers.

H(n) = (n+1)! × Σ 1/k!/((n+1-k)!) for k=1 to n, connecting to the binomial coefficients.

Harmonic numbers have been computed to arbitrary precision. For n = 10^100, H(n) ≈ 230.2585093... using the asymptotic formula.

The Harmonic Number Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Harmonic Number Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.