HarmonicNumber
gives the n harmonic number
.
HarmonicNumber[n,r]
gives the harmonic number of order r.
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- The harmonic numbers are given by
with
.
- HarmonicNumber can be evaluated to arbitrary numerical precision.
- HarmonicNumber automatically threads over lists.
Examples
open allclose allBasic Examples (7)
Plot over a subset of the integers:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Scope (25)
Numerical Evaluation (4)
Specific Values (4)
HarmonicNumber[n,a] for symbolic a:
HarmonicNumber[n,a] for symbolic n:
Find a value of n for which HarmonicNumber[n]=1.5:
Visualization (3)
Plot the HarmonicNumber function:
Plot the HarmonicNumber function for various orders:
Plot the real part of HarmonicNumber:
Plot the imaginary part of HarmonicNumber:
Function Properties (4)
Real domain of HarmonicNumber:
HarmonicNumber threads elementwise over lists and arrays:
TraditionalForm formatting:
Differentiation (3)
Integration (3)
Compute the indefinite integral using Integrate:
Series Expansions (2)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
General term in the series expansion using SeriesCoefficient:
Function Identities and Simplifications (2)
Generalizations & Extensions (5)
Harmonic Numbers (2)
Applications (4)
The average number of comparisons in Quicksort:
Book stacking with the maximal overhang:
Picking the best candidate out of n after x evaluated choices [more info]:
Properties & Relations (9)
Use FullSimplify to simplify expressions containing harmonic numbers:
Generate from sums and integrals:
HarmonicNumber can be represented as a DifferenceRoot:
General term in the series expansion of HarmonicNumber:
The generating function for HarmonicNumber:
The exponential generating function for HarmonicNumber:
Possible Issues (3)
Large arguments can give results too large to be computed explicitly:

Machine-number inputs can give high‐precision results:
Often results are expressed in PolyGamma instead of HarmonicNumber:
Text
Wolfram Research (1999), HarmonicNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/HarmonicNumber.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1999. "HarmonicNumber." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HarmonicNumber.html.
APA
Wolfram Language. (1999). HarmonicNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HarmonicNumber.html