gives the 
alternating harmonic number ![TemplateBox[{n}, AlternatingHarmonicNumber]](Files/AlternatingHarmonicNumber.en/67.png "TemplateBox[{n}, AlternatingHarmonicNumber]"){kind=small}
.
AlternatingHarmonicNumber[n,r]
gives the 
alternating harmonic number ![TemplateBox[{n, r}, AlternatingHarmonicNumber2]](Files/AlternatingHarmonicNumber.en/70.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2]"){kind=small}
of order 
.
AlternatingHarmonicNumber[n,r,s]
gives the 
alternating harmonic number ](Files/AlternatingHarmonicNumber.en/74.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2](s)"){kind=small}
of order 
and decoration value 
.
AlternatingHarmonicNumber
gives the 
alternating harmonic number ![TemplateBox[{n}, AlternatingHarmonicNumber]](Files/AlternatingHarmonicNumber.en/3.png "TemplateBox[{n}, AlternatingHarmonicNumber]"){kind=small}
.
AlternatingHarmonicNumber[n,r]
gives the 
alternating harmonic number ![TemplateBox[{n, r}, AlternatingHarmonicNumber2]](Files/AlternatingHarmonicNumber.en/6.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2]"){kind=small}
of order 
.
AlternatingHarmonicNumber[n,r,s]
gives the 
alternating harmonic number ](Files/AlternatingHarmonicNumber.en/10.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2](s)"){kind=small}
of order 
and decoration value 
.
Details
- AlternatingHarmonicNumber is the alternating analog of HarmonicNumber represented by sign–oscillating reciprocal sums.
- AlternatingHarmonicNumber appears in fields such as combinatorics, number theory and quantum field theory.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For positive integers
, one-argument AlternatingHarmonicNumber is given by ![TemplateBox[{n}, AlternatingHarmonicNumber]=sum_(i=1)^(n)(-1)^(i+1)/i](Files/AlternatingHarmonicNumber.en/14.png "TemplateBox[{n}, AlternatingHarmonicNumber]=sum_(i=1)^(n)(-1)^(i+1)/i")
. - For arbitrary
, ![TemplateBox[{n}, AlternatingHarmonicNumber]](Files/AlternatingHarmonicNumber.en/16.png "TemplateBox[{n}, AlternatingHarmonicNumber]")
admits an analytic continuation given by Log[2]-(-1)n LerchPhi[-1,1,1+n]. - For positive integers
, two-argument AlternatingHarmonicNumber is given by ![TemplateBox[{n, r}, AlternatingHarmonicNumber2]=sum_(i=1)^(n)(-1)^(i+1)/i^r](Files/AlternatingHarmonicNumber.en/18.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2]=sum_(i=1)^(n)(-1)^(i+1)/i^r")
. - For arbitrary
, ![TemplateBox[{n, r}, AlternatingHarmonicNumber2]](Files/AlternatingHarmonicNumber.en/20.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2]")
admits an analytic continuation given by 2-r ((-2+2r) Zeta[r]-(-1)n Zeta[r,
]+(-1)n Zeta[r,
]). - For positive integers
, three-argument AlternatingHarmonicNumber is given by =sum_(i=1)^(n)(-1)^(i+1)s^i/i^r](Files/AlternatingHarmonicNumber.en/24.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2](s)=sum_(i=1)^(n)(-1)^(i+1)s^i/i^r")
. - For arbitrary
, ](Files/AlternatingHarmonicNumber.en/26.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2](s)")
admits an analytic continuation given by (-s)1+n LerchPhi[-s,r,1+n]-PolyLog[r,-s]. - The alternating harmonic series
converges to 
. - The alternating harmonic series
converges to ![(1-2^(1-r)) TemplateBox[{r}, Zeta]=TemplateBox[{r}, DirichletEta]](Files/AlternatingHarmonicNumber.en/30.png "(1-2^(1-r)) TemplateBox[{r}, Zeta]=TemplateBox[{r}, DirichletEta]")
. - The alternating harmonic series
converges to ![-TemplateBox[{r, {-, s}}, PolyLog]](Files/AlternatingHarmonicNumber.en/32.png "-TemplateBox[{r, {-, s}}, PolyLog]")
. - AlternatingHarmonicNumber can be evaluated to arbitrary numerical precision.
- AlternatingHarmonicNumber automatically threads over lists.
- Sum, RSolve, GeneratingFunction, ExponentialGeneratingFunction, ZTransform and DirichletTransform include methods for handling inputs involving AlternatingHarmonicNumber.
Examples
open all close allBasic Examples (4)
First 10 alternating harmonic numbers:
Table[AlternatingHarmonicNumber[n], {n, 10}]Plot over a subset of the reals:
ReImPlot[AlternatingHarmonicNumber[t], {t, 0, 5}]Plot over a subset of the complexes:
ComplexPlot3D[AlternatingHarmonicNumber[z], {z, -1 - I, 1 + I}, ...]Series expansion at the origin:
Series[AlternatingHarmonicNumber[x], {x, 0, 3}]Scope (22)
Numerical Evaluation (6)
AlternatingHarmonicNumber[4, 1 / 2]N[AlternatingHarmonicNumber[3], 50]N[AlternatingHarmonicNumber[3, 5], 50]N[AlternatingHarmonicNumber[3, 5, 1 / 2], 50]AlternatingHarmonicNumber[4, 1 + I]N[AlternatingHarmonicNumber[27 + I, 5]]N[AlternatingHarmonicNumber[2 + 1 / 2I, 5 - I, I]]Evaluate efficiently at high precision:
AlternatingHarmonicNumber[13, 5`100]//TimingAlternatingHarmonicNumber[5 / 7, 5`1000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
AlternatingHarmonicNumber[2, Interval[{2.1, 2.2}]]AlternatingHarmonicNumber[1 / 2, CenteredInterval[3 / 4, 1 / 1000]]Or compute average-case statistical intervals using Around:
AlternatingHarmonicNumber[ 3 / 2, Around[2.1, 0.01]]Compute the elementwise values of an array:
AlternatingHarmonicNumber[2, {{1 / 2, 2}, {2, 1 / 2}}]Or compute the matrix AlternatingHarmonicNumber function using MatrixFunction:
MatrixFunction[AlternatingHarmonicNumber[2, #]&, {{1 / 2, 2}, {2, 1 / 2}}]//FullSimplifySpecific Values (6)
AlternatingHarmonicNumber[n,r] for symbolic 
:
AlternatingHarmonicNumber[4, r]AlternatingHarmonicNumber[n,r] for symbolic 
:
AlternatingHarmonicNumber[n, 4]//FunctionExpandAlternatingHarmonicNumber[0]
AlternatingHarmonicNumber[∞]AlternatingHarmonicNumber[∞, r]AlternatingHarmonicNumber[∞, r, s]Find a value of 
for which AlternatingHarmonicNumber[n]=0.5:
nval = Chop[n /. FindRoot[AlternatingHarmonicNumber[n] == 0.5, {n, 1}], 10^-6]//QuietPlot[Abs[AlternatingHarmonicNumber[n]], {n, 0, 5}, Rule[...]]Express alternating harmonic numbers of fractional arguments in terms of elementary functions:
AlternatingHarmonicNumber[1 / 2]//FunctionExpandAlternatingHarmonicNumber[7 / 3]//FunctionExpandFunction Properties (7)
Real domain of AlternatingHarmonicNumber:
FunctionDomain[AlternatingHarmonicNumber[x], x]FunctionDomain[AlternatingHarmonicNumber[z], z, Complexes]Real range of AlternatingHarmonicNumber:
FunctionRange[AlternatingHarmonicNumber[x], x, y]//QuietAlternatingHarmonicNumber is not an analytic function:
FunctionAnalytic[AlternatingHarmonicNumber[x], x]AlternatingHarmonicNumber is neither non-negative nor non-positive:
FunctionSign[AlternatingHarmonicNumber[x], x]AlternatingHarmonicNumber has no singularities and discontinuities in the real line:
FunctionSingularities[AlternatingHarmonicNumber[x], x]FunctionDiscontinuities[AlternatingHarmonicNumber[x], x]AlternatingHarmonicNumber is neither convex nor concave:
FunctionConvexity[AlternatingHarmonicNumber[x], x]TraditionalForm formatting:
TraditionalForm[{AlternatingHarmonicNumber[n], AlternatingHarmonicNumber[n, r], AlternatingHarmonicNumber[n, r, s]}]Series Expansions (3)
Find the Taylor expansion using Series:
Series[AlternatingHarmonicNumber[n], {n, 0, 3}]Plots of the first three approximations around 
:
terms = Normal@Table[Series[AlternatingHarmonicNumber[n], {n, 0, m}], {m, 1, 5, 2}];
ReImPlot[{AlternatingHarmonicNumber[n], terms}, {n, 0, 10}, PlotRange -> {-30, 30}]The series expansion at the unity:
Series[AlternatingHarmonicNumber[n], {n, 1, 3}]The series expansion at infinity:
Series[AlternatingHarmonicNumber[n], {n, ∞, 5}]Applications (10)
Summation (3)
A finite Euler sum involving an AlternatingHarmonicNumber:
Sum[n^2AlternatingHarmonicNumber[n, 3], {n, 1, k}]An infinite Euler sum involving an AlternatingHarmonicNumber:
Sum[((1/2))^n(AlternatingHarmonicNumber[n, 3]/n^4), {n, 1, ∞}]A finite Euler sum involving a product of a HarmonicNumber and an AlternatingHarmonicNumber:
Sum[(HarmonicNumber[n, 2]AlternatingHarmonicNumber[n]/3^nn^2), {n, 1, ∞}]Recurrence Equations (2)
Solve a recurrent equation containing AlternatingHarmonicNumber in the forcing term:
RSolveValue[{x[k] - 2x[k - 1] == AlternatingHarmonicNumber[k], x[1] == 1}, x[k], k]A product of a HarmonicNumber and an AlternatingHarmonicNumber in the forcing term, along with a pre-factor 
:
RSolveValue[{x[k] + 6x[k - 1] == k * HarmonicNumber[k] * AlternatingHarmonicNumber[k], x[1] == 1}, x[k], k]Generating Function (3)
Compute the GeneratingFunction of an AlternatingHarmonicNumber:
GeneratingFunction[AlternatingHarmonicNumber[n, 2], n, x]Compute the GeneratingFunction of the product of a HarmonicNumber and an AlternatingHarmonicNumber:
GeneratingFunction[HarmonicNumber[n]AlternatingHarmonicNumber[n, 3], n, x]Compute the ExponentialGeneratingFunction of the product of a StirlingS1 and an AlternatingHarmonicNumber:
ExponentialGeneratingFunction[StirlingS1[n, 3]AlternatingHarmonicNumber[n, 2], n, x]Discrete Transforms (2)
Compute the ZTransform of the square of an AlternatingHarmonicNumber:
ZTransform[AlternatingHarmonicNumber[n, 2] ^ 2, n, z]Compute the DirichletTransform of an AlternatingHarmonicNumber (with a rational pre-factor):
DirichletTransform[(1 / 2)^nAlternatingHarmonicNumber[n, 2], n, s]Properties & Relations (9)
The defining identities for AlternatingHarmonicNumber:
Sum[((-1)^i + 1/i), {i, 1, n}] == AlternatingHarmonicNumber[n]Sum[((-1)^i + 1/i^r), {i, 1, n}] == AlternatingHarmonicNumber[n, r]Sum[((-1)^i + 1s^i/i^r), {i, 1, n}] == AlternatingHarmonicNumber[n, r, s]![TemplateBox[{n, r}, AlternatingHarmonicNumber2]](Files/AlternatingHarmonicNumber.en/39.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2]"){kind=small}
![TemplateBox[{n}, AlternatingHarmonicNumber]](Files/AlternatingHarmonicNumber.en/40.png "TemplateBox[{n}, AlternatingHarmonicNumber]"){kind=small}
AlternatingHarmonicNumber[n, 1] === AlternatingHarmonicNumber[n]](Files/AlternatingHarmonicNumber.en/42.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2](s)"){kind=small}
![TemplateBox[{n, r}, AlternatingHarmonicNumber2]](Files/AlternatingHarmonicNumber.en/43.png "TemplateBox[{n, r}, AlternatingHarmonicNumber2]"){kind=small}
AlternatingHarmonicNumber[n, r, 1] === AlternatingHarmonicNumber[n, r]For integer 
, AlternatingHarmonicNumber can be expressed using a HarmonicNumber:
(AlternatingHarmonicNumber[n] == HarmonicNumber[n] - HarmonicNumber[Floor[n / 2]]) /. n -> 10AlternatingHarmonicNumber can be expressed using a decorated HarmonicNumber:
AlternatingHarmonicNumber[n] === -HarmonicNumber[n, 1, -1]AlternatingHarmonicNumber[n, r] === -HarmonicNumber[n, r, -1]AlternatingHarmonicNumber can be expressed using a decorated MultipleHarmonicNumber:
AlternatingHarmonicNumber[n] === -MultipleHarmonicNumber[n, {1}, {-1}]AlternatingHarmonicNumber[n, r] === -MultipleHarmonicNumber[n, {r}, {-1}](AlternatingHarmonicNumber[z + 1] == AlternatingHarmonicNumber[z] + ((-1)^z/Sqrt[(1 + z)^2]))//FunctionExpand//FullSimplifyRelations between AlternatingHarmonicNumber, LerchPhi and Zeta functions:
AlternatingHarmonicNumber[n] == -(-1)^n LerchPhi[-1, 1, 1 + n] + Log[2]//FullSimplifyAlternatingHarmonicNumber[n, r] == 2^-r ((-2 + 2^r) Zeta[r] + (-1)^n (-Zeta[r, (1 + n/2)] + Zeta[r, (2 + n/2)]))//FullSimplifyAlternatingHarmonicNumber[n, r, s] == (-s)^1 + n LerchPhi[-s, r, 1 + n] - PolyLog[r, -s]//FullSimplifyAlternatingHarmonicNumber at 
can be expressed using Zeta, DirichletEta and PolyLog:
AlternatingHarmonicNumber[∞, r] == (1 - 2^-r + 1)Zeta[r] == DirichletEta[r]//FullSimplifyAlternatingHarmonicNumber[∞, r, s] == -PolyLog[r, -s]//FullSimplifyDefine Fermat quotient 
and the utility modZeroQ to test whether a rational expr is 
mod 
:
fermatQuotient[a_, p_Integer ? PrimeQ] := (a ^ (p - 1) - 1) / p;modZeroQ[expr_, m_Integer] := Module[{t = Together[expr], num, den}, num = Numerator[t];den = Denominator[t];
If[CoprimeQ[den, m], Mod[num * PowerMod[den, -1, m], m] == 0, False]
];Verify the mod-p congruence ![TemplateBox[{{p, -, 1}}, AlternatingHarmonicNumber] = 2 q_p(2) (mod p)](Files/AlternatingHarmonicNumber.en/50.png "TemplateBox[{{p, -, 1}}, AlternatingHarmonicNumber] = 2 q_p(2) (mod p)"){kind=small}
across many primes:
AllTrue[Prime[Range[100]], modZeroQ[AlternatingHarmonicNumber[# - 1] - 2 fermatQuotient[2, #], #]&]Verify the mod-p2 congruence ![TemplateBox[{{p, -, 1}}, AlternatingHarmonicNumber] = 2 q_p(2) -p q_p(2)^2 (mod p^2)](Files/AlternatingHarmonicNumber.en/51.png "TemplateBox[{{p, -, 1}}, AlternatingHarmonicNumber] = 2 q_p(2) -p q_p(2)^2 (mod p^2)"){kind=small}
(for 
):
AllTrue[Prime[Range[3, 100]], modZeroQ[AlternatingHarmonicNumber[# - 1] - (2 fermatQuotient[2, #] - # fermatQuotient[2, #] ^ 2), # ^ 2]&]Text
Wolfram Research (2026), AlternatingHarmonicNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/AlternatingHarmonicNumber.html.
CMS
Wolfram Language. 2026. "AlternatingHarmonicNumber." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AlternatingHarmonicNumber.html.
APA
Wolfram Language. (2026). AlternatingHarmonicNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AlternatingHarmonicNumber.html