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Alternating Harmonic Number

The nth alternating harmonic number is the number obtained by taking alternate signs in the sum defining the harmonic number H_n,

H_n^'=sum_(k=1)^(n)((-1)^(k+1))/k
(1)
=ln2+1/2(-1)^n[psi_0(1/2n+1/2)-psi_0(1/2n+1)]
(2)
=ln2+1/2(-1)^n[H_((n-1)/2)-H_(n/2)],
(3)

where psi_0(z) is the digamma function. The even-indexed alternating harmonic numbers have the form

H_(2n)^'=sum_(k=1)^(2n)((-1)^(k+1))/k
(4)
=sum_(k=1,3,...)^(2n)((-1)^(k+1))/k+sum_(k=2,4,...)^(2n)((-1)^(k+1))/k
(5)
=sum_(k=1,3,...)^(2n)1/k-sum_(k=2,4,...)^(2n)1/k
(6)
=(sum_(k=1,3,...)^(2n)1/k+sum_(k=2,4,...)^(2n)1/k)-2sum_(k=2,4,...)^(2n)1/k
(7)
=sum_(k=1)^(2n)1/k-sum_(k=1)^(n)1/k
(8)
=H_(2n)-H_n.
(9)

Alternating harmonic numbers are implemented in the Wolfram Language as AlternatingHarmonicNumber[n].

See also

Alternating Harmonic Series, Harmonic Number

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References

Li, R. "Integrals of Polylogarithms and Infinite Series Involving Generalized Harmonic Numbers." 19 Mar 2021. https://arxiv.org/abs/2103.12590.

Cite this as:

Weisstein, Eric W. "Alternating Harmonic Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlternatingHarmonicNumber.html

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