Shapiro's Cyclic Sum Constant

Author

Eric W. Weisstein
April 16, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/ShapirosCyclicSumConstant.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plot

Plot[{Exp[-x],2/(Exp[x]+Exp[x/2])},{x,-.1,.1},PlotStyle->{Red,Blue}]

Plot[Max[Exp[-x],2/(Exp[x]+Exp[x/2])],{x,-.001,.001},PlotStyle->Red,AspectRatio->Automatic]

Limit[(Exp[0]-Exp[h])/h,h->0]

Limit[(2/(Exp[h]+Exp[h/2])-(2/(Exp[0]+Exp[0/2])))/h,h->0]

Simplify[(Exp[x]-2/(Exp[x]+Exp[x/2]))/(2x)]

Plot[(((E^x)-(2/((E^(x/2))+(E^x))))/(2 x)),{x,-2,2}]

Simplify[D[(((E^x)-(2/((E^(x/2))+(E^x))))/(2 x)),x]]

FindRoot[((-(E^x)+(2/((E^(x/2))+(E^x)))+((E^x)+((2+(E^(-x/2)))/(((1+(E^(x/2))))^2))) x)/(2 (x^2)))==0,{x,-.1,-1}]

1-Exp[-0.6814447114794688`]

Limit[%,x->0]

N[%]

2/(Exp[x]+Exp[x/2])/.x->.2

Exp[.1]

D[Exp[-x],x]

Series[2/(Exp[x]+Exp[x/2]),{x,0,2}]

s1=Normal[Series[Exp[-x],{x,ϵ,2}]]

Limit[%,ϵ->0]

s2=Simplify[Normal[Series[2/(Exp[x]+Exp[x/2]),{x,-ϵ,2}]]]

Simplify[(s1+s2)/2]

Limit[%,ϵ->0]

Plot[(1/2) (2-((7 x)/4)+((3 (x^2))/4)-((41 (x^3))/192)),{x,-.01,.01}]

V7

?FindInstance

FindInstance[((-(E^x)+(2/((E^(x/2))+(E^x)))+((E^x)+((2+(E^(-x/2)))/(((1+(E^(x/2))))^2))) x)/(2 (x^2)))==0,x]//FullSimplify

1-Exp[Root[{(((E^(-#1)) ((Sech[((#1)/4)])^2) (2+#1+(E^((#1)/2)) (((((E^((#1)/2))+(E^(#1))))^2) (-1+#1)+2 (1+#1))))/(4 ((#1)^2)))&,-0.6814447113581229913538071359934990739675880171640627380381`30.}]]//FullSimplify

N[%]