Logarithmic Series

Author

Eric W. Weisstein
October 26, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/LogarithmicSeries.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LogarithmicSeries.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Sum

(Underoverscript[(OverHat[∑]), (k=1), ∞])ln(k)==(1/2) ln(2 π)

Numerical

NSum[Log[k],{k,∞}]

SymbolicSum

SymbolicSum`SymbolicSum[Log[k],{k,∞},GenerateConditions->False]//FullSimplify

v4.2.1

Sum[Log[k],{k,∞}]

Sum[(-1)^k Log[k]/k,{k,∞}]

FullSimplify[Sum[Log[k],{k,∞}]]

v5.0

Sum[Log[k],{k,∞}]

Sum[(-1)^k Log[k]/k,{k,∞}]

Sum[Log[k],{k,∞}]

v5.1

Sum[Log[k],{k,∞}]

V7

Sum[Log[k],{k,∞}]

Sum[Log[k],{k,∞},Regularization->"Dirichlet"]

(Underoverscript[(OverHat[∑]), (k=1), ∞])(((-1))^k) ln(k)==(1/2) ln((π/2))

Numerical

NSum[(-1)^k Log[k],{k,∞}]

N[(1/2) Log[(π/2)]]

v4.2

SymbolicSum`SymbolicSum[(-1)^k Log[k],{k,∞},GenerateConditions->False]//FullSimplify

FullSimplify[Sum[(-1)^k Log[k],{k,∞}]]

v5.0

Sum[(-1)^k Log[k],{k,∞}]

v5.1

SymbolicSum`SymbolicSum[(-1)^k Log[k],{k,∞},GenerateConditions->False]//FullSimplify

Sum[(-1)^k Log[k],{k,∞}]

V7

Sum[(-1)^k Log[k],{k,∞}]

Sum[(-1)^k Log[k],{k,∞},Regularization->"Dirichlet"]

(∑_(k=1)^∞)(((((-1))^k) ln(k))/k)==ℽ ln(2)-(((ln^2)(2))/2)

NSum[(-1)^k Log[k]/k,{k,∞}]

v4.2

SymbolicSum`SymbolicSum[(-1)^k Log[k]/k,{k,∞},GenerateConditions->False]//FullSimplify

FullSimplify[Sum[(-1)^k Log[k]/k,{k,∞}]]

v5.0

SymbolicSum`SymbolicSum[(-1)^k Log[k]/k,{k,∞},GenerateConditions->False]//FullSimplify

Sum[(-1)^k Log[k]/k,{k,∞}]

v5.1

SymbolicSum`SymbolicSum[(-1)^k Log[k]/k,{k,∞},GenerateConditions->False]//FullSimplify

Sum[(-1)^k Log[k]/k,{k,∞}]

V7

Sum[(-1)^k Log[k]/k,{k,∞}]//FullSimplify

(∑_(k=1)^∞)(((((-1))^k) ln(k))/(k^n))==(2^(-n)) (ln(4) (Zeta[n])+((2^n)-2) ζ'(n))

v4.2

FullSimplify[Sum[(-1)^k Log[k]/k^n,{k,∞}]]

v5.0

FullSimplify[Sum[(-1)^k Log[k]/k^n,{k,∞}]]

FullSimplify[Log[2] Zeta[n]+(2^(-n)) (-2+(2^n)) Zeta'[n]-(2^(-n)) (Log[4] Zeta[n]+(-2+(2^n)) Zeta'[n])]

v5.1

Sum[(-1)^k Log[k]/k^n,{k,∞}]

FullSimplify[Sum[(-1)^k Log[k]/k^n,{k,∞}]]

v7

Sum[(-1)^k Log[k]/k^n,{k,∞}]//FullSimplify