Natural Logarithm of 2

Author

Eric W. Weisstein
February 8, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/NaturalLogarithmof2.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/NaturalLogarithmof2.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Properties

Element[Log[2],Algebraics]

Decimal Digits

N[Log[2],21]

Prepend[First[RealDigits[Log[2],10,20]],0]

Sums

(lim_(n->∞)) (Underoverscript[(1/n)∑, k, n])(((((-1))^(k+1)) n)/k)

(1/n)Sum[(((-1))^(k+1))(n/k),{k,n}]

Limit[%,n->∞]

(∑_k^∞)((((-1))^(k-1))/k)

Mercator Series

Sum[((((-1))^(k-1))/k),{k,∞}]

(1/2) (∑_(k=0)^∞)(1/((2^k) (k+1)))

(1/2)Sum[(1/(2^k))(1/(k+1)),{k,0,∞}]//FullSimplify

(∑_k^∞)(1/(k (2^k)))

Sum[(1/(k (2^k))),{k,∞}]

(2/3) (∑_(k=0)^∞)(1/((9^k) (2 k+1)))

(2/3)Sum[(1/((9^k)(2k+1))),{k,0,∞}]

%//FullSimplify

(3/4)-(1/8)(∑_k^∞)(((((-1))^(k-1)) (5 k+1) (Binomial[2 k, k]))/((16^k) (k (k+(1/2)))))

Lupas

V6

Sum[(((-1))^k)(((((Pochhammer[x,k])/(Pochhammer[x+2y,k]))))^2),{k,0,∞}]/.{x->1,y->1}

Apart//@Solve[y==-4 (-3+4ln2),ln2]

(3/4)-(1/8)Sum[(((-1))^(k-1))(1/(16^k))Binomial[2k,k]((5k+1)/(k(k+(1/2)))),{k,∞}]//N

(3/4)-(1/8)Sum[(((-1))^(k-1))(1/(16^k))Binomial[2k,k]((5k+1)/(k(k+(1/2)))),{k,∞}]//FullSimplify

FullSimplify[TrigToExp[%]]

V7

(3/4)-(1/8)Sum[(((-1))^(k-1))(1/(16^k))Binomial[2k,k]((5k+1)/(k(k+(1/2)))),{k,∞}]

FullSimplify[%]

2 (∑_(j=1)^n)((((-1))^(j+1))/(2 j))+(1/8) (((-1))^n) (2 n-2)! (∑_(k=0)^∞)(1/(16^k))(-(4/(Subscript[(8 k+3), 2 n-1]))-(4/(Subscript[(8 k+4), 2 n-1]))-(2/(Subscript[(8 k+5), 2 n-1]))+(1/(Subscript[(8 k+7), 2 n-1]))+(1/(Subscript[(8 k+8), 2 n-1]))+(8/(Subscript[(8 k+1), 2 n-1])))

Cloitre

This should give ln 2, but doesn't quite...

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

2 (∑_(j=1)^n)((((-1))^(j+1))/(2 j))+(1/8) (((-1))^n) (2 n-2)! (∑_(k=0)^∞)(1/(16^k))(-(4/(Subscript[(8 k+3), 2 n-1]))-(4/(Subscript[(8 k+4), 2 n-1]))-(2/(Subscript[(8 k+5), 2 n-1]))+(1/(Subscript[(8 k+7), 2 n-1]))+(1/(Subscript[(8 k+8), 2 n-1]))+(8/(Subscript[(8 k+1), 2 n-1])))

2*Sum[(-1)^(j + 1)/(2*j), {j, 1, n}] + (1/8)*(-1)^n*(2*n - 2)!*
   Sum[(1/16^k)*(-(4/Pochhammer[8*k + 3, 2*n - 1]) - 
      4/Pochhammer[8*k + 4, 2*n - 1] - 
      2/Pochhammer[8*k + 5, 2*n - 1] + 
      1/Pochhammer[8*k + 7, 2*n - 1] + 
      1/Pochhammer[8*k + 8, 2*n - 1] + 
      8/Pochhammer[8*k + 1, 2*n - 1]), {k, 0, Infinity}]

TraditionalForm[sum=Sum[((((-1))^(j+1))/j),{j,n}]+(((-1))^n)(((2n-2)!)/8)Sum[(1/(16^k))((8/(Pochhammer[8k+1,2n-1]))-(4/(Pochhammer[8k+3,2n-1]))-(4/(Pochhammer[8k+4,2n-1]))-(2/(Pochhammer[8k+5,2n-1]))+(1/(Pochhammer[8k+7,2n-1]))+(1/(Pochhammer[8k+8,2n-1]))),{k,0,∞}]]

FullSimplify[Table[sum,{n,6}]]

...not quite right...

6 (∑_n^∞)(1/(((e^(π n))-1) n))+10 (∑_n^∞)(1/(((e^(π n))+1) n))-4 (∑_n^∞)(1/(((e^(2 π n))+1) n))==log(2)

10Sum[(1/(n(Exp[π n]+1))),{n,∞}]+6Sum[(1/(n(Exp[π n]-1))),{n,∞}]-4Sum[(1/(n(Exp[2π n]+1))),{n,∞}]

N[%,20]

Log[2]//N

Continued Fraction

ContinuedFraction[Log[2],100]

ContinuedFraction[Log[2]^2,100]

Plot of Truncated Series

<<Utilities`Plot`

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

IntegerListPlot[Table[(1/2) (((-1))^n) (HarmonicNumber[(1/2) (-1+n)]-HarmonicNumber[(n/2)])+Log[2],{n,50}],
AxesLabel->TraditionalForm/@{n,TraditionalForm[HoldForm[Sum[((((-1))^(k+1))/k),{k,n}]]]},
Epilog->{Blue,Line[{{0,#},{50,#}}]&[Log[2]]}]

Truncated Series

V6

<<Utilities`Simplify`

LogExpand[Sum[((((-1))^(k-1))/k),{k,N}]]//Simplify

FullSimplify[%]

Series[(1/2) (PolyGamma[0,1+(N/2)]-PolyGamma[0,((1+N)/2)]),{N,∞,10}]

Series[Tan[x],{x,0,10}]

Sum[(((((-1))^(n-1))(2^(2n))((2^(2n))-1)BernoulliB[2n])/((2n)!))(x^(2n-1)),{n,0,5}]

{1,2,16,272,7936}Table[((((-1))^k)/((4^k)(N^(2k)))),{k,5}]

Sum[((((-1))^(k+1))/k),{k,n}]

ps[n_]:=((Log[4])/2)-(1/2) (((-1))^n) (PolyGamma[0,1+(n/2)]-PolyGamma[0,((1+n)/2)])

Log[2.]

s="0."<>StringJoin@@ToString/@(Transpose[RealDigits[#,10,62][[1]]&/@{ps[N[5 10^6,1000]]//Chop,Log[2]}]/.{{x_,x_}:>x,{x_,y_}:>"*"})

{e1,e2,e3}=Chop/@N[{ps[N[5 10^6,1000]],Log[2],s},62]

pos=First/@StringPosition[s,"*"]

p={1,0}+#&/@Partition[Prepend[Last/@Split[pos,#2-#1==1&]-2,0],2,1]

diffs=ToString/@(-Subtract@@@ToExpression/@Function[pos,StringTake[#,pos]&/@(StringDrop[#,2]&/@ToString/@Chop/@{e1,e2})]/@p)

e3=StringJoin[Sequence@@Table[" ",{62}]];
Do[e3=StringReplacePart[e3,diffs[[i]],{p[[i]][[2]]-StringLength[diffs[[i]]]+3,2+p[[i]][[2]]}],{i,Length[diffs]}];
e3

Column[{e1,e2,e3}]

V7

Sum[((((-1))^(k-1))/k),{k,N}]

Sum[(((((-1))^(n-1))(2^(2n))((2^(2n))-1)BernoulliB[2n])/((2n)!))(x^(2n-1)),{n,0,∞}]//InputForm

Series[Sum[(((((-1))^(n-1))(2^(2n))((2^(2n))-1)BernoulliB[2n])/((2n)!))(x^(2n-1)),{n,0,20}],{x,0,10}]

Series[Tan[x],{x,0,10}]

SeriesCoefficient[Tan[x],{x,0,n}]

Log[2]+(((-1))^n)((1/(2n))+Sum[((((-1))^k)/\[Placeholder])

-((I (((2 I))^n) (-1+(((-1))^n)) (-1+(2^(1+n))) BernoulliB[1+n])/((1+n)!))/.n->n-1

Table[-((I (((2 I))^(-1+n)) (-1+(((-1))^(-1+n))) (-1+(2^n)) BernoulliB[n])/(n!)),{n,6}]

FullSimplify[-((I (((2 I))^(-1+n)) (-1+(((-1))^(-1+n))) (-1+(2^n)) BernoulliB[n])/(n!)),n∈Integers&&n>0]

Table[(((-1))^(n-1))(2^(2n))((2^(2n))-1)BernoulliB[2n]/(2n),{n,10}]

FullSimplify[(((-1))^(n-1))(2^(2n))((2^(2n))-1)BernoulliB[2n]/(2n),n∈Integers&&n>0]

-(((((-1))^n) (2^(-1+2 n)) (-1+(4^n)) BernoulliB[2 n])/n)/.n->k

Sum[-(((((-1))^k) (2^(-1+2 k)) (-1+(4^k)) BernoulliB[2 k])/k)((((-1))^k)/((4^k)(n^(2k)))),{k,0,N}]

BBP-Type Formula

BBP

<<Utilities`Simplify`

FindBasis[Log[2],(-1),1]

BasisSum[#,k]&/@%

ReleaseHold[%]

Simplify[%]

Search

<<Utilities`Simplify`

Monitor[Do[If[(b=FindBasis[Log[2],-6^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-6^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-5^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-5^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,1}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[2],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{3^k,n,p}];Print["DONE"]

Special

(1/35917)((230166911/9240)-(∑_k^∞)((((1/2)))^k) ((9/(k+2))+(8/(k+3))+(7/(k+4))+(6/(k+5))-(6/(k+7))-(7/(k+8))-(8/(k+9))-(9/(k+10))-(10/(k+11))+(11/k)+(10/(k+1))))==log(2)

(230166911/9240-NSum[(1/2)^k*(11/k+10/(k+1)+9/(k+2)+8/(k+3)+7/(k+4)+6/(k+5)-6/(k+7)-7/(k+8)-8/(k+9)-9/(k+10)-10/(k+11)),{k,∞}])/35917

(230166911/9240-Sum[(1/2)^k*(11/k+10/(k+1)+9/(k+2)+8/(k+3)+7/(k+4)+6/(k+5)-6/(k+7)-7/(k+8)-8/(k+9)-9/(k+10)-10/(k+11)),{k,∞}])/35917//Simplify

Summary

log(2)

BasisSum[#,k]&/@{
{Log[2],{3,0,-2,-1,0,0,-3},-27,6,1},
{Log[2],{3,-9,-8,-1,1,0,6},-27,6,1},{Log[2],{-2187,0,1458,729,0,0,81,0,-54,-27,0,0,-3,0,2,1,0,0,2187},-19683,18,1},
{Log[2],{2187,-6561,-5832,-729,729,0,-81,243,216,27,-27,0,3,-9,-8,-1,1,0,4374},-19683,18,1},
{Log[2],{1594323,0,-1062882,-531441,0,0,-59049,0,39366,19683,0,0,2187,0,-1458,-729,0,0,-81,0,54,27,0,0,3,0,-2,-1,0,0,-1594323},-14348907,30,1},
{Log[2],{1594323,-4782969,-4251528,-531441,531441,0,-59049,177147,157464,19683,-19683,0,2187,-6561,-5832,-729,729,0,-81,243,216,27,-27,0,3,-9,-8,-1,1,0,3188646},-14348907,30,1}
}//ColumnForm//TraditionalForm

BasisSum[#,k]&/@{
{Log[2],{-2,0,1,1,2},-4,4,1},
{Log[2],{8,-4,-1,-8},-8,3,1},
{Log[2],{0,16,32,8,0,8,4,-2,0,1,-1,-1,-32},-64,12,1},{Log[2],{-32,16,0,-8,8,8,0,2,4,1,0,1,32},-64,12,1},
{Log[2],{512,-256,-64,-64,32,8,8,-4,-1,-512},-512,9,1},{Log[2],{512,0,-256,-256,-128,0,64,64,32,0,-16,-16,-8,0,4,4,2,0,-1,-1,-512},-1024,20,1},{Log[2],{0,-768,256,-384,0,192,-64,96,0,32,16,-24,0,12,16,6,0,-3,1,1,512},-1024,20,1},{Log[2],{8192,0,-4096,-4096,-2048,0,1024,1024,512,0,-256,-256,-128,0,64,64,32,0,-16,-16,-8,0,4,4,2,0,-1,-1,-8192},-16384,28,1},{Log[2],{32768,-16384,-4096,-4096,2048,512,512,-256,-64,-64,32,8,8,-4,-1,-32768},-32768,15,1}
}//ColumnForm//TraditionalForm

BasisSum[#,k]&/@Flatten[{{{Log[2],{-1,1},-1,1,1}},{{Log[2],{-1,1,-1,1},-1,3,1}},{{Log[2],{-1,1,-1,1,-1,1},-1,5,1}},{{Log[2],{-1,1,-1,1,-1,1,-1,1},-1,7,1}},{{Log[2],{-1,1,-1,1,-1,1,-1,1,-1,1},-1,9,1}},{{Log[2],{-1,1,-1,1,-1,1,-1,1,-1,1,-1,1},-1,11,1}},{{Log[2],{-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1},-1,13,1}},{{Log[2],{-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1},-1,15,1}}},1]//ColumnForm//TraditionalForm

BasisSum[#,k]&/@{
{Log[2],{1,-2},2,1,1},{Log[2],{2,1,-4},4,2,1},{Log[2],{4,2,1,-8},8,3,1},{Log[2],{8,4,2,1,-16},16,4,1},{Log[2],{8,0,-4,-4,-2,0,1,1,-8},16,8,1},{Log[2],{16,8,4,2,1,-32},32,5,1},{Log[2],{-32,-16,-8,-4,-2,-1,64},64,6,1},{Log[2],{64,32,16,8,4,2,1,-128},128,7,1},{Log[2],{128,64,32,16,8,4,2,1,-256},256,8,1},{Log[2],{128,0,-64,-64,-32,0,16,16,8,0,-4,-4,-2,0,1,1,-128},256,16,1},{Log[2],{256,128,64,32,16,8,4,2,1,-512},512,9,1},{Log[2],{-512,-256,-128,-64,-32,-16,-8,-4,-2,-1,1024},1024,10,1},{Log[2],{1024,512,256,128,64,32,16,8,4,2,1,-2048},2048,11,1},{Log[2],{-2048,-1024,-512,-256,-128,-64,-32,-16,-8,-4,-2,-1,4096},4096,12,1},{Log[2],{768,768,256,512,-64,192,64,160,-64,112,0,16,-12,16,-4,-2,1,5,-1,3,1,2,0,1,-2048},4096,24,1},{Log[2],{4096,2048,1024,512,256,128,64,32,16,8,4,2,1,-8192},8192,13,1},{Log[2],{8192,4096,2048,1024,512,256,128,64,32,16,8,4,2,1,-16384},16384,14,1},{Log[2],{16384,8192,4096,2048,1024,512,256,128,64,32,16,8,4,2,1,-32768},32768,15,1},
{Log[2],{32768,16384,8192,4096,2048,1024,512,256,128,64,32,16,8,4,2,1,-65536},65536,16,1},
{Log[2],{32768,0,-16384,-16384,-8192,0,4096,4096,2048,0,-1024,-1024,-512,0,256,256,128,0,-64,-64,-32,0,16,16,8,0,-4,-4,-2,0,1,1,-32768},65536,32,1}
}//ColumnForm//TraditionalForm

BasisSum[#,k]&/@{
{Log[2],{-2,0,3},9,2,1},{Log[2],{-18,0,-2,0,27},81,4,1},{Log[2],{162,0,18,0,2,0,-243},729,6,1},{Log[2],{-81,0,54,27,0,0,3,0,-2,-1,0,0,81},729,12,1},{Log[2],{-81,324,54,27,0,36,3,0,-2,3,0,0,-162},729,12,1},{Log[2],{1458,0,162,0,18,0,2,0,-2187},6561,8,1},{Log[2],{13122,0,1458,0,162,0,18,0,2,0,-19683},59049,10,1},{Log[2],{118098,0,13122,0,1458,0,162,0,18,0,2,0,-177147},531441,12,1},{Log[2],{59049,0,-39366,-19683,0,0,-2187,0,1458,729,0,0,81,0,-54,-27,0,0,-3,0,2,1,0,0,-59049},531441,24,1},{Log[2],{-59049,177147,157464,19683,-19683,0,2187,-6561,-5832,-729,729,0,-81,243,216,27,-27,0,3,-9,-8,-1,1,0,-118098},531441,24,1},{Log[2],{1062882,0,118098,0,13122,0,1458,0,162,0,18,0,2,0,-1594323},4782969,14,1},{Log[2],{-9565938,0,-1062882,0,-118098,0,-13122,0,-1458,0,-162,0,-18,0,-2,0,14348907},43046721,16,1}
}//ColumnForm//TraditionalForm

(log^2)(2)

b=FindBasis[Log[2]^2,2^6,6,BasisPower->2,WorkingPrecision->300]

BasisSum[#,k]&/@b//TraditionalForm

ReleaseHold[%4]/.Utilities`Simplify`Private`HeldSum:>Sum

FullSimplify[%]

N[%,20]

Sum Limit

(lim_(n->∞)) ((∑_(k=(2^n))^((2^(n+1))-1))(1/k))

v5

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

Limit[%,n->∞]

N[-PolyGamma[0,(2^n)]+PolyGamma[0,(2^(1+n))]/.n->10^7,20]

v6

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

Limit[%,n->∞]

N[-PolyGamma[0,(2^n)]+PolyGamma[0,(2^(1+n))]/.n->10^7,20]

V7

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

Limit[%,n->∞]

Integrals

Integrate[(1/(x ((Log[x])^2))),{x,2,∞}]