Harmonic number
The nth harmonic number, denoted H_n, is defined as the partial sum of the first n terms of the harmonic series:H_n = \sum_{k=1}^n \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}. [1] This sequence begins with H_1 = 1, H_2 = 1.5, H_3 \approx 1.833, and increases without bound as n grows, though the increments become progressively smaller.[2] Although the infinite harmonic series \sum_{k=1}^\infty \frac{1}{k} diverges to infinity, the partial sums grow very slowly, approximately logarithmically—much more slowly than the linear divergence of the geometric series with ratio 1. A key asymptotic approximation for large n is H_n \approx \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + \cdots, where \gamma \approx 0.57721 is the Euler-Mascheroni constant, defined as the limit \gamma = \lim_{n \to \infty} (H_n - \ln n).[2] This approximation arises from integral comparisons, where H_n bounds the natural logarithm from above and below.[3] The concept of harmonic numbers traces its origins to ancient mathematical traditions, including studies of harmonic proportions in geometry and music theory, with systematic development in the 17th and 18th centuries by figures like Jacob Bernoulli, who analyzed the series' divergence.[4] They play a fundamental role in diverse areas of mathematics, including number theory (e.g., connections to the prime harmonic series and zeta functions), analysis (e.g., in integral tests and Stirling's approximation for factorials), and combinatorics (e.g., identities involving binomial coefficients and permutations).[5][6]
Definition and Fundamentals
Definition
The nth harmonic number, denoted H_n, is defined as the partial sum of the first n terms of the harmonic series: H_n = \sum_{k=1}^n \frac{1}{k} for each positive integer n.[7] These numbers are named after the harmonic series, a concept rooted in the musical theory of overtones, where the frequencies of sound waves form integer multiples of a fundamental tone, leading to the reciprocal proportions in the series.[4] The explicit study of these partial sums as a distinct sequence traces back to Jacob Bernoulli, who in 1689 examined the harmonic series in his Tractatus de Seriebus Infinitis, later appended to his posthumously published Ars Conjectandi (1713); there, Bernoulli provided a proof of the series' divergence, highlighting the growth of the partial sums H_n.[8] Unlike the infinite harmonic series \sum_{k=1}^\infty \frac{1}{k}, which diverges to infinity, the harmonic numbers H_n remain finite for any finite n but increase without bound as n grows.[8] For example, the first harmonic number is H_1 = 1, serving as the initial term from which subsequent values build additively.[7]Notation and Conventions
The nth harmonic number is conventionally denoted by H_n, where n is a non-negative integer, representing the partial sum \sum_{k=1}^n \frac{1}{k} of the harmonic series.[7] This notation is standard in modern mathematical literature and emphasizes the cumulative nature of the sum up to the index n.[9] By convention, the zeroth harmonic number is defined as H_0 = 0, corresponding to the empty sum before any terms are included.[1] The indexing typically begins at n = [1](/page/1), where H_[1](/page/1) = [1](/page/1), and increases thereafter, with the partial sum explicitly written to clarify the range when necessary.[10] Signed variants of harmonic numbers introduce alternating signs in the sum. The basic signed harmonic number is given by the partial sum \sum_{k=1}^n \frac{(-1)^{k+1}}{k}, which forms the alternating harmonic series and converges to \ln 2 in the limit.[11] This form is often referred to simply as the alternating harmonic number in the literature, without a unique superscripted notation, though it relates to the Dirichlet eta function for broader generalizations. For more general forms, the notation H_n^{(s)} is used to denote the generalized harmonic number \sum_{k=1}^n \frac{1}{k^s} for positive integers s \geq 1, extending the standard case where s = 1.[7] In contemporary usage, LaTeX and similar typesetting systems standardize H_n with subscript indexing, ensuring clarity and consistency across texts, while older literature occasionally employed varied symbols that have largely been supplanted by this convention.[12]Initial Values and Computation
The harmonic numbers for small indices can be obtained by direct summation of the reciprocals of the positive integers, providing concrete examples that illustrate their rational nature. For instance, H_1 = 1 = \frac{1}{1}, H_2 = 1 + \frac{1}{2} = \frac{3}{2}, H_3 = \frac{3}{2} + \frac{1}{3} = \frac{11}{6}, H_4 = \frac{11}{6} + \frac{1}{4} = \frac{25}{12}. [1] The table below lists the first ten harmonic numbers in reduced fractional form, computed via successive addition:| n | H_n |
|---|---|
| 1 | $1/1 |
| 2 | $3/2 |
| 3 | $11/6 |
| 4 | $25/12 |
| 5 | $137/60 |
| 6 | $49/20 |
| 7 | $363/140 |
| 8 | $761/280 |
| 9 | 7129/2520 |
| 10 | 7381/2520 |
Approximations and Asymptotics
Logarithmic Approximation
The nth harmonic number H_n admits a fundamental asymptotic approximation given byH_n \approx \ln n + \gamma,
where \gamma \approx 0.5772156649 is the Euler-Mascheroni constant. This relation captures the leading-order growth of H_n, reflecting the divergent nature of the harmonic series in a logarithmic scale. The constant \gamma is rigorously defined as
\gamma = \lim_{n \to \infty} (H_n - \ln n),
a limit first established by Leonhard Euler in his analysis of harmonic progressions.[15] The approximation derives from integral comparisons that bound the discrete sum defining H_n. Consider the function f(x) = 1/x, which is decreasing for x > 0. By integral test remainder estimates, the sum satisfies
\int_1^{n+1} \frac{1}{x} \, dx < H_n < 1 + \int_1^n \frac{1}{x} \, dx,
or equivalently,
\ln(n+1) < H_n < 1 + \ln n.
These inequalities imply that H_n - \ln n is bounded between \ln(n+1) - \ln n = \ln(1 + 1/n) and 1, and monotonicity arguments show the difference converges to the finite limit \gamma as n \to \infty.[16] A straightforward error bound quantifies the accuracy of the leading approximation:
|H_n - \ln n - \gamma| < \frac{1}{2n}.
This estimate follows from the next-order term in the asymptotic expansion being approximately $1/(2n), with higher-order corrections ensuring the remainder is smaller in magnitude for sufficiently large n.[16]