Knuth's Series

Author

Eric W. Weisstein
October 5, 2008

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

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

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

Sum

(∑_k^∞)(((k^k)/((e^k) k!))-(1/(Sqrt[2 π k])))==-((Zeta[(1/2)])/(Sqrt[2 π]))-(2/3)

v5.2

Sum[((k^k)/(k! (E^k)))-(1/(Sqrt[2π k])),{k,∞}]

v6

Sum[((k^k)/(k! (E^k)))-(1/(Sqrt[2π k])),{k,∞}]

V7

Sum[((k^k)/(k! (E^k)))-(1/(Sqrt[2π k])),{k,∞}]

(∑_k^∞)((1/(Sqrt[2 π k]))-((Pochhammer[(1/2), k-1])/((Sqrt[2]) (k-1)!)))==((Zeta[(1/2)])/(Sqrt[2 π]))

Plot

Plot[(1/(Sqrt[2π k]))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),{k,1,10},PlotStyle->Red]

Exact

(1/(Sqrt[2π]))Zeta[1/2]//N

V5.2

Sum[(1/(Sqrt[2π k]))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),{k,∞}]

NSum[(1/(Sqrt[2π k]))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),{k,∞}]

NSum[(1/(Sqrt[2π k]))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),{k,10^4}]//Chop//Timing

V6

$Version

Sum[(1/(Sqrt[2π k]))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),{k,∞}]

V7

$Version

Sum[1/Sqrt[2*Pi*k]-Pochhammer[1/2,k-1]/((k-1)!*Sqrt[2]),{k,Infinity}]

Sum[(1/(Sqrt[2π k]))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),{k,1000}]//N

(∑_k^∞)((1/(Sqrt[2 π k]))-(((2 k-3)!!)/((2^(k-1)) ((Sqrt[2]) (k-1)!))))==((Zeta[(1/2)])/(Sqrt[2 π]))

Table[Pochhammer[1/2,k-1],{k,5}]

Table[(2n-3)!!/2^(n-1),{n,5}]

V5.2

Sum[(1/(Sqrt[2π k]))-(((2k-3)!!/2^(k-1))/((k-1)!(Sqrt[2]))),{k,∞}]

Expand[FullSimplify[(1/(Sqrt[2π k]))-(((2k-3)!!/2^(k-1))/((k-1)!(Sqrt[2]))),k∈Integers&&k>0]]

a1=Table[((Gamma[k-(1/2)])/((Sqrt[π]) Gamma[k])),{k,10}]

a2=Table[(2J-3)!!/(2J-2)!!,{J,10}]

(1/(Sqrt[2]))NSum[(1/(Sqrt[π k]))-(((2k-3)!!)/((2k-2)!!)),{k,∞}]

V6

$Version

Sum[(1/(Sqrt[2π k]))-(((2k-3)!!/2^(k-1))/((k-1)!(Sqrt[2]))),{k,∞}]

(1/(Sqrt[2]))NSum[(1/(Sqrt[π k]))-(((2k-3)!!)/((2k-2)!!)),{k,∞}]

V7

$Version

Sum[(1/(Sqrt[2π k]))-(((2k-3)!!/2^(k-1))/((k-1)!(Sqrt[2]))),{k,∞}]

(∑_k^∞)(((Gamma[k])-(Sqrt[k]) (Gamma[k-(1/2)]))/((Sqrt[2 π]) (Sqrt[k]) (Gamma[k])))==((Zeta[(1/2)])/(Sqrt[2 π]))

v5.2

FullSimplify[(1/(Sqrt[2π k]))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),k∈Integers&&k>0]

NSum[((-(Sqrt[k]) Gamma[-(1/2)+k]+Gamma[k])/((Sqrt[k]) (Sqrt[2 π]) Gamma[k])),{k,∞}]

NSum[((-(Sqrt[k]) Gamma[-(1/2)+k]+Gamma[k])/((Sqrt[k]) (Sqrt[2 π]) Gamma[k])),{k,10^3}]

Sum[((k^k)/(k! (E^k)))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),{k,∞}]

FullSimplify[FunctionExpand[((k^k)/(k! (E^k)))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2])))]]

Sum[-((Gamma[-(1/2)+k])/((Sqrt[2 π]) Gamma[k]))+(((E^(-k)) (k^k))/(Gamma[1+k])),{k,∞}]

(((-Pi)/(x Sin[π x]))(1/(Gamma[x]))/.x->1/2-k)/.Gamma[1/2-k]:>(-1)^k 2^k/(2k-1)!!(Sqrt[π])

Simplify[Csc[((1/2)-k) π],k∈Integers]

Simplify[-(((((-2))^(-k)) (Sqrt[π]) (((-1))^k) (-1+2 k)!!)/((1/2)-k))]

With[{k=3},(((2^(1-k)) (Sqrt[π]) (-1+2 k)!!)/(-1+2 k))]//N

Gamma[3-1/2]//N

FullSimplify[(-((((2^(1-k)) (Sqrt[π]) (-1+2 k)!!)/(-1+2 k))/((Sqrt[2 π]) Gamma[k]))+(((E^(-k)) (k^k))/(Gamma[1+k]))),k∈Integers]

Sum[(((Sqrt[(2/π)]) Gamma[(1/2)+k])/(Gamma[k]-2 Gamma[1+k]))+(((E^(-k)) (k^k))/(Gamma[1+k])),{k,∞}]

v6

FullSimplify[(1/(Sqrt[2π k]))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2]))),k∈Integers&&k>0]

NSum[((-(Sqrt[k]) Gamma[-(1/2)+k]+Gamma[k])/((Sqrt[k]) (Sqrt[2 π]) Gamma[k])),{k,∞}]

$Version

Sum[((-(Sqrt[k]) Gamma[-(1/2)+k]+Gamma[k])/((Sqrt[k]) (Sqrt[2 π]) Gamma[k])),{k,∞}]

FullSimplify[FunctionExpand[((k^k)/(k! (E^k)))-((Pochhammer[1/2,k-1])/((k-1)!(Sqrt[2])))]]

$Version

Sum[-((Gamma[-(1/2)+k])/((Sqrt[2 π]) Gamma[k]))+(((E^(-k)) (k^k))/(Gamma[1+k])),{k,∞}]

Sum[-((Gamma[-(1/2)+k])/((Sqrt[2 π]) Gamma[k]))+(((E^(-k)) (k^k))/(Gamma[1+k])),{k,2000}]//N

N[Zeta[1/2]/Sqrt[2*Pi]]

Sum[(((Sqrt[(2/π)]) Gamma[(1/2)+k])/(Gamma[k]-2 Gamma[1+k]))+(((E^(-k)) (k^k))/(Gamma[1+k])),{k,∞}]

V7

$Version

Sum[((-(Sqrt[k]) Gamma[-(1/2)+k]+Gamma[k])/((Sqrt[k]) (Sqrt[2 π]) Gamma[k])),{k,∞}]

Limit

From Bailey et al. 2006, p. 221.  But something's wrong with it.

FullSimplify[D[ProductLog[-z/E],z]+1/(2-2z)]

Limit[%,z->1]

Supposedly 2/3...

Plot[(1/(2-2 z))+((ProductLog[-(z/E)])/(z+z ProductLog[-(z/E)])),{z,.9,1.},PlotStyle->Red]

Series Expansion of Term

Series[((k^k)/(k! (E^k)))-(1/(Sqrt[2π k])),{k,∞,5}]

Attempt to Find Closed Form

Apart[Simplify[Normal[Series[z!/(Sqrt[2π]),{z,∞,7}]],z>0]Exp[z](z^(-1/2-z))]

Series[((k^k)/(k! (E^k)))-(1/(Sqrt[2π k])),{k,∞,7}]

Normal[((Series[z!,{z,∞,7}]Exp[z](z^(-1/2-z))-(Sqrt[2π]))/(2π)/.z->-z)(z^(-1/2))]

NSum[((k^k)/(k! (E^k)))-(1/(Sqrt[2π k])),{k,∞}]//Timing

NSum[((k^k)/(k! (E^k)))-(1/(Sqrt[2π k])),{k,∞},WorkingPrecision->10]//Timing

N[-(2/3)-(1/(Sqrt[2π]))Zeta[1/2],20]

SeriesCoefficient[((k^k)/(k! (E^k)))-(1/(Sqrt[2π k])),{k,∞,n}]