FoxTrot Series

Author

Eric W. Weisstein
August 11, 2008

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

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

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

Sum

V6

Sum[(((((-1))^(n+1))(n^2))/((n^3)+1)),{n,∞}]//FullSimplify//TraditionalForm

Factor[(((((-1))^(n+1))(n^2))/((n^3)+1))]

Apart[-(( (n^2))/((1+n) (1-n+(n^2))))]

s1=Sum[-((((-1))^n)/(3 (1+n))),{n,∞}]

FullSimplify[%]

Times@@(n-#&/@(List@@Last/@Roots[1-n+(n^2)==0,n]))

Apart[(1/((-(((-1))^(1/3))+n) ((((-1))^(2/3))+n)))]

(1/3)Sum[(((((-1))^(2/3))/((1+(((-1))^(1/3))) ((((-1))^(1/3))-n)))+((((-1))^(2/3))/((1+(((-1))^(1/3))) ((((-1))^(2/3))+n))))(((-1))^n) (1-2n),{n,∞}]

s2=FullSimplify[%]

FullSimplify[s1+s2]

N[s=(1/3) (1-Log[2]+π Sech[(((Sqrt[3]) π)/2)]),20]

NSum[(((((-1))^(n+1))(n^2))/((n^3)+1)),{n,∞}]

Sum[(((((-1))^(n+1))(n^2))/((n^3)+1)),{n,∞}]//FullSimplify

N[%]//Chop

FullSimplify[(1/6) (2-Log[4]-PolyGamma[0,(1/2) (((-1))^(1/3))]-PolyGamma[0,-(1/2) (((-1))^(2/3))]+PolyGamma[0,(1/2) (1+(((-1))^(1/3)))]+PolyGamma[0,(1/2) (1-(((-1))^(2/3)))])-(1/3) (1-Log[2]+π Sech[(((Sqrt[3]) π)/2)])]

Developer`ZeroQ[%]

V7

Sum[(((((-1))^(n+1))(n^2))/((n^3)+1)),{n,∞}]