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Harmonic number

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: H n = 1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k . {\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}.}

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Overview

Identities involving harmonic numbers

Calculation

Generating functions

Arithmetic properties

Applications

Generalizations

Harmonic numbers for real and complex values

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Harmonic number

Nodes79
Edges78
Triples111
Avg. degree1.97
Density0.025316
Components1

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Harmonic number

Top relations

related to References · 46
Harmonic number → Addison-Wesley, Adv, Apery Numbers, Appl, Archived, Arthur, Basel, Benjamin, Beukers' Conjecture, Binomial Coefficient Identity Associated, Carsten, Cite, CiteSeerX, Combinatorics, Computer Programming, Computer Proofs, Donald Knuth, Ed Sandifer, Estimating, Fundamental Algorithms
related to Arithmetic properties · 11
Harmonic number → As, Eisenstein, Fermat, Furthermore, Indeed, It, More, The, Theisinger, Wieferich, Wolstenholme's
related to Roman harmonic numbers · 10
Harmonic number → Closed, Daniel Loeb, Donald Knuth, Gian-Carlo Rota, If, Of, Steven Roman, Stirling, The Roman, There
related to Hyperharmonic numbers · 8
Harmonic number → Conway, Guy, In, Let, Numbers, The, The Book, Then
related to External links · 7
Harmonic number → Creative Commons Attribution/Share-Alike License, Eric, Harmonic, MathWorld, PlanetMath, This, Weisstein
related to Harmonic numbers for real and complex values · 6
Harmonic number → Euler, Mascheroni, Riemann, Taylor, The, The Taylor
has application · 5
Harmonic number → In, Jeffrey Lagarias, Riemann, The, This
related to Generating functions · 4
Harmonic number → An, Ein, Gamma, The
related to Multiplication formulas · 4
Harmonic number → For, Riemann, The, Using
related to Identities involving harmonic numbers · 3
Harmonic number → By, Stirling, The

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Important terminology

displaystyle harmonic numbers frac number sum function infty zeta left right -1 also riemann int begin aligned end ln series

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Harmonic numberis asum of the reciprocals of the first n natural numbers0.90text
Harmonic numberhas applicationThe0.60section
Harmonic numberhas applicationThis0.60section
Harmonic numberhas applicationIn0.60section
Harmonic numberhas applicationJeffrey Lagarias0.60section
Harmonic numberhas applicationRiemann0.60section
Harmonic numberrelated to Arithmetic propertiesThe0.60section
Harmonic numberrelated to Arithmetic propertiesIt0.60section
Harmonic numberrelated to Arithmetic propertiesTheisinger0.60section
Harmonic numberrelated to Arithmetic propertiesIndeed0.60section
Harmonic numberrelated to Arithmetic propertiesMore0.60section
Harmonic numberrelated to Arithmetic propertiesAs0.60section

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