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Logarithmic Series

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Infinite series of various simple functions of the logarithm include

sum_(k=1)^^^inftylnk=1/2ln(2pi)
(1)
sum_(k=1)^^^infty(-1)^klnk=1/2ln(1/2pi)
(2)
sum_(k=1)^(infty)((-1)^klnk)/k=gammaln2-1/2(ln2)^2
(3)
sum_(k=1)^(infty)((-1)^klnk)/(k^n)=(ln2zeta(n)+(2^(n-1)-1)zeta^'(n))/(2^(n-1)),
(4)

where gamma is the Euler-Mascheroni constant and zeta(z) is the Riemann zeta function. Note that the first two of these are divergent in the classical sense, but converge when interpreted as zeta-regularized sums.

See also

Logarithm, False Logarithmic Series, Zeta-Regularized Sum

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References

Bromwich, T. J. I'A. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 351, 1991.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 37, 1999.

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Logarithmic Series

Cite this as:

Weisstein, Eric W. "Logarithmic Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogarithmicSeries.html

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