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HarmonicNumber
✖
HarmonicNumber

✖

gives the n^(th) harmonic number TemplateBox[{n}, HarmonicNumber].

✖

gives the harmonic number TemplateBox[{n, r}, HarmonicNumber2] of order r.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For positive integers n, the harmonic numbers are given by TemplateBox[{n, r}, HarmonicNumber2]=sum_(i=1)^(n)1/i^r with TemplateBox[{n}, HarmonicNumber]=TemplateBox[{n, 1}, HarmonicNumber2].
  • For arbitrary n and r≠1, the numerical value of TemplateBox[{n, r}, HarmonicNumber2] is given by Zeta[r]-HurwitzZeta[r,n+1].
  • HarmonicNumber can be evaluated to arbitrary numerical precision.
  • HarmonicNumber automatically threads over lists.
  • HarmonicNumber can be used with Interval and CenteredInterval objects. »

Examples

open allclose all

Basic Examples  (7)Summary of the most common use cases

First ten harmonic numbers:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-hl13l
Out[1]=1

Plot over a subset of the integers:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-25h1s
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-kiedlx
Out[1]=1

Series expansion at the origin:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-fdkkja
Out[1]=1

Series expansion at Infinity:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-20imb
Out[1]=1

Series expansion at a singular point:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-d2klx1
Out[1]=1

Carry out sums involving harmonic numbers:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-b57d70
Out[1]=1

Scope  (35)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-l274ju
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-wlv0g
Out[2]=2

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-b0wt9
Out[1]=1

The precision of the output tracks the precision of the input:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-y7k4a
Out[2]=2

Complex number input:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-bq2c6r
Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-h0d6g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-emr3q7
Out[2]=2

Or compute average-case statistical intervals using Around:

In[3]:=3
✖
https://wolfram.com/xid/09cdorq2a-cw18bq
Out[3]=3

Compute the elementwise values of an array:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-thgd2
Out[1]=1

Or compute the matrix HarmonicNumber function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-o5jpo
Out[2]=2

Specific Values  (5)

HarmonicNumber[n,a] for symbolic a:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-fc9m8o
Out[1]=1

HarmonicNumber[n,a] for symbolic n:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-h7p5ce
Out[1]=1

Value at zero:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-bmqd0y
Out[1]=1

Find a value of n for which HarmonicNumber[n]=1.5:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-f2hrld
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-bm9h1
Out[2]=2

Express harmonic numbers of fractional arguments in terms of elementary functions:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-nk52mh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-n4ycy
Out[2]=2

Visualization  (3)

Plot the HarmonicNumber function:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-b1j98m
Out[1]=1

Plot the HarmonicNumber function for various orders:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-ecj8m7
Out[1]=1

Plot the real part of HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-b8vnf
Out[1]=1

Plot the imaginary part of HarmonicNumber:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-fjwo7k
Out[2]=2

Function Properties  (11)

Real domain of HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-cl7ele
Out[1]=1

Complex domain:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-de3irc
Out[2]=2

Real range of HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-5j1gt2
Out[1]=1

HarmonicNumber threads elementwise over lists and arrays:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-g4bqu6
Out[1]=1

HarmonicNumber is not an analytic function:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-h5x4l2
Out[1]=1

However, it is meromorphic:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-1snzk8
Out[2]=2

HarmonicNumber is neither non-increasing nor non-decreasing:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-g6kynf
Out[1]=1

HarmonicNumber is not injective:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-fkbstt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-9fxpr3
Out[2]=2

HarmonicNumber is surjective:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-hkqec4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-hdm869
Out[2]=2

HarmonicNumber is neither non-negative nor non-positive:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-84dui
Out[1]=1

HarmonicNumber has both singularities and discontinuities for negative integers:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-mdtl3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-mn5jws
Out[2]=2

HarmonicNumber is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-kdss3
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-l5d69d

Differentiation  (3)

First derivative with respect to n:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-krpoah
Out[1]=1

Higher derivatives with respect to n:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-z33jv
Out[1]=1

Plot the higher derivatives with respect to n:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-fxwmfc
Out[2]=2

Formula for the ^(th) derivative with respect to n:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-cb5zgj
Out[1]=1

Integration  (3)

Compute the indefinite integral using Integrate:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-bponid
Out[1]=1

Verify the anti-derivative:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-op9yly
Out[2]=2

Definite integral:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-b9jw7l
Out[1]=1

More integrals:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-4nbst
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-h9t4zs
Out[2]=2

Series Expansions  (2)

Find the Taylor expansion using Series:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-ewr1h8
Out[1]=1

Plots of the first three approximations around :

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-binhar
Out[2]=2

General term in the series expansion using SeriesCoefficient:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-dznx2j
Out[1]=1

Function Identities and Simplifications  (2)

The defining identity for HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-b5js3b
Out[1]=1

Recurrence identities:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-d34lia
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-liy2m1
Out[2]=2

Generalizations & Extensions  (5)Generalized and extended use cases

Harmonic Numbers  (2)

Series expansion at infinity:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-chlfqs
Out[1]=1

HarmonicNumber can be applied to power series:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-ejq61x
Out[1]=1

Harmonic Numbers of Order r  (3)

Evaluate at exact arguments:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-def9vx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-jj7vas
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09cdorq2a-89i7j
Out[3]=3

Series expansion at any point:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-vp64w
Out[1]=1

Series expansion at infinity:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-gfa8uu
Out[1]=1

Applications  (5)Sample problems that can be solved with this function

The average number of comparisons in Quicksort:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-d1sgx2
Out[1]=1

Plot over the complex plane:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-jyr60y
Out[1]=1

Book stacking with the maximal overhang:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-cguq1w
Out[1]=1

Pick the best candidate out of n candidates after x evaluated choices [more info]:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-h7f0w

Evaluate for n=100:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-dh9nl7
Out[2]=2

Plot as a function of the size of the candidate pool:

In[3]:=3
✖
https://wolfram.com/xid/09cdorq2a-kri504
Out[3]=3

A finite sum with StirlingS1 expressed in terms of HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-b5uguq
Out[1]=1

A finite sum with StirlingS2 expressed in terms of HarmonicNumber:

In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-dco1yi
Out[2]=2

Properties & Relations  (10)Properties of the function, and connections to other functions

HarmonicNumber can be expressed in terms of PolyGamma:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-dqx8pq
Out[1]=1

HarmonicNumber can be expressed in terms of Zeta and HurwitzZeta:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-kmgk7z
Out[1]=1

Use FullSimplify to simplify expressions containing harmonic numbers:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-3mmk2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-bh6ykg
Out[2]=2

Expand in simpler functions:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-gvy9r2
Out[1]=1

Sums:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-c5syve
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-xl7aq
Out[2]=2

Generate from sums and integrals:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-ckuacd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-esv1y7
Out[2]=2

HarmonicNumber can be represented as a DifferenceRoot:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-ojcks5
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-d7mw27
Out[2]=2

General term in the series expansion of HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-dlm2b6
Out[1]=1

The ordinary generating function for HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-pz93yz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-frz0l8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09cdorq2a-nalti6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/09cdorq2a-e1gujp
Out[4]=4

The exponential generating function for HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-gaiyeu
Out[1]=1

Possible Issues  (3)Common pitfalls and unexpected behavior

Large arguments can give results too large to be computed explicitly:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-cg5joy
Out[1]=1

Machine-number inputs can give high‐precision results:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-bwfqfq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-bmtdv
Out[2]=2

Results are often expressed in terms of PolyGamma instead of HarmonicNumber:

In[1]:=1
✖
https://wolfram.com/xid/09cdorq2a-c1jr3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09cdorq2a-e7zau2
Out[2]=2
Wolfram Research (1999), HarmonicNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/HarmonicNumber.html.
✖
Wolfram Research (1999), HarmonicNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/HarmonicNumber.html.

Text

Wolfram Research (1999), HarmonicNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/HarmonicNumber.html.

✖
Wolfram Research (1999), HarmonicNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/HarmonicNumber.html.

CMS

Wolfram Language. 1999. "HarmonicNumber." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HarmonicNumber.html.

✖
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

✖
Wolfram Language. (1999). HarmonicNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HarmonicNumber.html

BibTeX

@misc{reference.wolfram_2025_harmonicnumber, author="Wolfram Research", title="{HarmonicNumber}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/HarmonicNumber.html}", note=[Accessed: 25-March-2025 ]}

✖
@misc{reference.wolfram_2025_harmonicnumber, author="Wolfram Research", title="{HarmonicNumber}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/HarmonicNumber.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_harmonicnumber, organization={Wolfram Research}, title={HarmonicNumber}, year={1999}, url={https://reference.wolfram.com/language/ref/HarmonicNumber.html}, note=[Accessed: 25-March-2025 ]}

✖
@online{reference.wolfram_2025_harmonicnumber, organization={Wolfram Research}, title={HarmonicNumber}, year={1999}, url={https://reference.wolfram.com/language/ref/HarmonicNumber.html}, note=[Accessed: 25-March-2025 ]}