Harmonic Number Calculator

What Is a Harmonic Number?

The nth harmonic number Hₙ adds up the reciprocals of the whole numbers from 1 to n:

Each harmonic number is the previous one plus one more term, so Hₙ = Hₙ₋₁ + 1/n. By convention H₀ = 0, because a sum with no terms is 0. The infinite sum 1 + 1/2 + 1/3 + … is called the harmonic series, and Hₙ is its nth partial sum.

The name comes from music: a string divided into halves, thirds and quarters gives the harmonics (overtones) of its note, and those lengths are 1/2, 1/3, 1/4 of the whole.

Table of Harmonic Numbers

The fractions get long quickly: H₁₀₀₀ written as a fraction has a 434-digit numerator. Decimals are rounded to 10 places, the same as the calculator.

Why the Harmonic Series Never Stops Growing

The terms 1/n shrink towards 0, yet their sum has no limit: the harmonic series diverges. The classic proof (Nicole Oresme, about 1350) groups the terms so each group is at least 1/2:

Adding infinitely many halves gives no finite total, so Hₙ passes every number eventually. It just does so very slowly:

• Hₙ first passes 2 at n = 4, 3 at n = 11, 4 at n = 31 and 5 at n = 83.
• It needs n = 12,367 terms to pass 10.
• Even after 10¹⁵ terms (a thousand trillion), Hₙ is only about 35.116.

The graph shows Hₙ for n = 1 to 20 (dots) next to the curve ln n + γ. The dots climb more slowly each step but never level off.

The ln n + γ Approximation

For large n, adding every term is slow. Harmonic numbers stay close to the natural logarithm plus a constant:

Hₙ is always a little above ln n + γ, and the gap is close to 1/(2n), so it shrinks as n grows:

Adding more correction terms makes the formula far more accurate. This is the version the calculator uses above n = 2,000:

Two simple bounds also hold for every n ≥ 1: ln(n + 1) < Hₙ ≤ 1 + ln n.

Properties of Harmonic Numbers

• Recurrence: Hₙ = Hₙ₋₁ + 1/n, starting from H₀ = 0.
• Never whole: H₁ = 1 is the only harmonic number from H₁ on that is a whole number. For n ≥ 2, Hₙ always has an even denominator.
• Sum of harmonic numbers: H₁ + H₂ + … + Hₙ = (n + 1)Hₙ − n. For n = 4 that is 5 × 25/12 − 4 = 77/12.
• Digamma link: Hₙ = ψ(n + 1) + γ, which extends harmonic numbers to non-whole n.
• Slow growth: doubling n adds about ln 2 ≈ 0.693 to Hₙ.

Worked Examples

Find H₄. Put every term over the common denominator 12:

Find H₁₀. Using the denominator 2520 (the smallest number divisible by 1 to 10) gives H₁₀ = 7381/2520 ≈ 2.9289682540. The approximation ln 10 + γ ≈ 2.8798007579 is 0.049 too low, close to 1/(2 × 10) = 0.05.

Find H₁₀₀₀ without adding 1,000 terms.

The exact value is 7.4854708606, so one correction term already gives 7 correct digits.

Where Harmonic Numbers Appear

• Coupon collector problem: collecting all n different coupons, picked at random, takes n × Hₙ tries on average. Rolling every face of a die takes 6 × H₆ = 14.7 rolls on average.
• Quicksort: sorting n items in random order takes 2(n + 1)Hₙ − 4n comparisons on average, which is about 1.39 n log₂ n.
• Stacking blocks: n identical blocks of length 1, one per level, can overhang a table edge by up to Hₙ ÷ 2. With 4 blocks the top block sticks out by 25/24, more than its whole length.
• Number theory: harmonic numbers are linked to the Riemann zeta function, and Hₙ ≈ ln n is why the average number of divisors of a number near n is about ln n.
• Records: in a random sequence of n values, the expected number of new highs (records) is Hₙ.

Harmonic Numbers in Excel and Code

Floating-point sums give 2.9289682539682538. Python's fractions.Fraction gives the exact 7381/2520. For huge n, use ln n + γ + 1/(2n) instead of a loop, or scipy.special.digamma(n + 1) + numpy.euler_gamma.

Common Mistakes

• Thinking the series converges. The terms go to 0, but the sum does not settle; that alone never proves convergence.
• Confusing it with a geometric series. 1 + 1/2 + 1/4 + 1/8 + … converges to 2; 1 + 1/2 + 1/3 + 1/4 + … does not.
• Leaving out the first term. The sum starts with 1/1 = 1.
• Adding the denominators. 1/2 + 1/3 is 5/6, not 2/5.
• Mixing up the harmonic mean. The harmonic mean of 1 to n is n ÷ Hₙ, not Hₙ itself.

Key Terms

Reciprocal

1 divided by a number; the reciprocal of 4 is 1/4.

Hₙ

The nth harmonic number, 1 + 1/2 + … + 1/n.

Series

The sum of the terms of a sequence.

Diverge

Grow without limit instead of settling on a value.

γ

The Euler-Mascheroni constant, about 0.5772156649.

ln n

The natural logarithm of n, to base e.

How to Use the Harmonic Number Calculator

1. Enter the number of terms. Type a whole number n from 0 up to 10¹⁵ (1,000,000,000,000,000). This is how many reciprocals 1, 1/2, 1/3, … are added together.
2. Read the result. The harmonic number Hₙ appears as you type, to 10 decimal places. The table below it shows the sum, the exact fraction for n up to 2,000, the approximation ln n + γ and the gap between Hₙ and that approximation.
3. Copy or start again. Press Copy to copy Hₙ, or Clear or Esc to start again. Negative numbers and decimals get a short message explaining why they do not work.

Example: enter 10 and the result is 2.9289682540. The table shows the fraction 7381/2520, ln 10 + γ ≈ 2.8798007579 and the gap 0.0491674961.

Who Can Use This Calculator?

Frequently Asked Questions (FAQs)