Harmonic Number — Definition, Formula & Examples

The

n

th harmonic number, written

H_n

, is the sum of the reciprocals of the first

n

positive integers:

1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}

. Harmonic numbers grow without bound as

n

increases, but they do so very slowly.

For each positive integer $n$ , the $n$ th harmonic number is defined as $H_n = \sum_{k=1}^{n} \frac{1}{k}$ . The sequence $\{H_n\}$ is strictly increasing and unbounded, meaning $\lim_{n \to \infty} H_n = \infty$ . For large $n$ , harmonic numbers satisfy the asymptotic relation $H_n \approx \ln n + \gamma$ , where $\gamma \approx 0.5772$ is the Euler–Mascheroni constant.

Key Formula

H_n = \sum_{k=1}^{n} \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}

Where:

$H_n$ = The $n$th harmonic number
$n$ = A positive integer indicating how many reciprocal terms to sum
$k$ = Index of summation running from 1 to $n$

How It Works

To compute

H_n

, simply add the fractions

\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}

. Each successive harmonic number builds on the previous one:

H_n = H_{n-1} + \frac{1}{n}

. Because each added term

\frac{1}{n}

is positive, the sequence is strictly increasing. However, the terms shrink toward zero, so growth is extremely slow — you need over

10^{43}

terms before

H_n

exceeds 100.

Worked Example

Problem: Compute the 5th harmonic number,

H_5

Write out the sum: List the reciprocals of the first 5 positive integers.

H_5 = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}

Find a common denominator: The least common denominator of 1, 2, 3, 4, and 5 is 60.

H_5 = \frac{60}{60} + \frac{30}{60} + \frac{20}{60} + \frac{15}{60} + \frac{12}{60}

Add the fractions: Sum the numerators over the common denominator.

H_5 = \frac{60 + 30 + 20 + 15 + 12}{60} = \frac{137}{60}

Answer:

H_5 = \dfrac{137}{60} \approx 2.2833

Why It Matters

Harmonic numbers appear throughout calculus and computer science. In algorithm analysis, the average number of comparisons in quicksort involves

H_n

. They also provide the classic proof that the harmonic series diverges, a foundational result in any study of infinite series.

Common Mistakes

Mistake: Assuming the harmonic series converges because its terms

\frac{1}{n}

approach zero.

Correction: A series can diverge even when its terms tend to zero. The harmonic series is the most famous example — use the comparison test or the integral test to confirm divergence.