e

Author

Eric W. Weisstein
April 19, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Hypergeometric identity

HypergeometricPFQ[{},{},1]

Integral Plot

V5

<<Graphics`InequalityGraphics`
<<Utilities`Typesetting`

Show[Block[{$DisplayFunction=Identity},
{
Plot[1/x,{x,.001,3},
Epilog->{Dashing[{.02}],Line[{{1,0},{1,1}}],Line[{{E,0},{E,1/E}}]}],
InequalityPlot[1<x<E&&0<y<1/x,{x,0,3},{y,0,2}]
}],AspectRatio->Automatic,PlotRange->{{0,3},{0,2}},
Ticks->{{1,2,{E,TraditionalForm[E]},3},
{#,NumberForm[#,{1,2},NumberPadding->{"","0"}]}&/@{0.5,1.0,1.5,2.}
},
AxesLabel->TraditionalForm/@{x,y==1/x}]

V6

Show[{
Plot[1/x,{x,.001,3},PlotRange->{{0,3},{0,2}},AxesLabel->{x,y==1/x}],
Plot[1/x,{x,1,E},Filling->0,FillingStyle->Hue[.5]]
},
AspectRatio->Automatic,Epilog->{Dashing[.02],Line[{{1,0},{1,1}}],Line[{{E,0},{E,1/E}}]},
Ticks->{{1,2,{E,e},3},
{#,NumberForm[#,{1,2},NumberPadding->{"","0"}]}&/@{0.5,1.0,1.5,2.}}]

Plot of Limit

Plot[(1+1/x)^x,{x,.1,200},PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,(1+1/x)^x},
Epilog->{Dashing[{.02}],Blue,Line[{{0,E},{200,E}}]}
]

Series

(∑_(k=0)^∞)(1/(k!))

Sum[1/k!,{k,0,∞}]

((((∑_(k=0)^∞)((((-1))^k)/(k!))))^(-1))

1/Sum[(-1)^k/k!,{k,0,∞}]

(∑_(k=0)^∞)((2 k+2)/((2 k+1)!))

V6

Sum[((2k+2)/((2k+1)!)),{k,0,∞}]

FullSimplify[%]

V7

Sum[((2k+2)/((2k+1)!)),{k,0,∞}]

FullSimplify[%]

V8

Sum[((2k+2)/((2k+1)!)),{k,0,∞}]

FullSimplify[%]

(∑_(k=0)^∞)((2 k+1)/((2 k)!))

V6

Sum[((2k+1)/((2k)!)),{k,0,∞}]

FullSimplify[%]

V7

Sum[((2k+1)/((2k)!)),{k,0,∞}]

FullSimplify[%]

V8

Sum[((2k+1)/((2k)!)),{k,0,∞}]

FullSimplify[%]

(1/2) (∑_(k=0)^∞)((k+1)/(k!))

(1/2)Sum[((k+1)/(k!)),{k,0,∞}]

2 (∑_(k=0)^∞)((k+1)/((2 k+1)!))

V6

2Sum[((k+1)/((2k+1)!)),{k,0,∞}]

FullSimplify[%]

FullSimplify[Sum[(1/(k!)),{k,0,∞,2}]+Sum[((k+3)/(2k!)),{k,1,∞,2}]]

V7

2Sum[((k+1)/((2k+1)!)),{k,0,∞}]

FullSimplify[%]

V8

2Sum[((k+1)/((2k+1)!)),{k,0,∞}]

FullSimplify[%]

Sum[(1/(k!)),{k,0,∞,2}]+Sum[((k+3)/(2 k!)),{k,1,∞,2}]==((3 e)/2)

Sum[(1/(k!)),{k,0,∞,2}]+Sum[((k+3)/(2k!)),{k,1,∞,2}]

FullSimplify[%]

((((∑_(k=0)^∞)((1-2 k)/((2 k)!))))^(-1))

V7

((Sum[((1-2k)/((2k)!)),{k,0,∞}])^(-1))

FullSimplify[%]

V8

((Sum[((1-2k)/((2k)!)),{k,0,∞}])^(-1))

FullSimplify[%]

(∑_(k=0)^∞)((3-4 (k^2))/((2 k+1)!))

V7

Sum[((3-4(k^2))/((2k+1)!)),{k,0,∞}]

FullSimplify[%]

V8

Sum[((3-4(k^2))/((2k+1)!)),{k,0,∞}]

FullSimplify[%]

(∑_(k=0)^∞)(((((3 k))^2)+1)/((3 k)!))

V7

Sum[(((((3k))^2)+1)/((3k)!)),{k,0,∞}]

FullSimplify[%]

V8

Sum[(((((3k))^2)+1)/((3k)!)),{k,0,∞}]

FullSimplify[%]

((((∑_(k=0)^∞)((4 k+3)/((2^(2 k+1)) (2 k+1)!))))^2)

V7

((Sum[((4k+3)/((2^(2k+1))(2k+1)!)),{k,0,∞}])^2)

FullSimplify[%]

V8

((Sum[((4k+3)/((2^(2k+1))(2k+1)!)),{k,0,∞}])^2)

FullSimplify[%]

3-(∑_k^∞)(1/(k (k+1) (k+1)!))==e

Christoph Reineke (pers. comm., Sep. 1, 2008 )

V7

3-Sum[(1/(k(k+1)(k+1)!)),{k,∞}]

Pippinger Products

(∏_n^∞)((((((((((2^(n-1))-1)!!))^2) ((((2^n)!!))^2))/(2 ((((2^(n-1))!!))^2) (((((2^n)-1)!!))^2)))))^(1/(2^n)))==(e/2)

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

Times@@Table[((((((((((2^(n-1))-1)!!))^2)((((2^n)!!))^2))/(2((((2^(n-1))!!))^2)(((((2^n)-1)!!))^2)))))^(1/(2^n))),{n,10}]//N

E/2.

TimeConstrained[NProduct[((((((((((2^(n-1))-1)!!))^2)((((2^n)!!))^2))/(2((((2^(n-1))!!))^2)(((((2^n)-1)!!))^2)))))^(1/(2^n))),{n,∞}],300]//Timing

TimeConstrained[Product[((((((((((2^(n-1))-1)!!))^2)((((2^n)!!))^2))/(2((((2^(n-1))!!))^2)(((((2^n)-1)!!))^2)))))^(1/(2^n))),{n,∞}],300]//Timing

exp(ℽ)==(Sqrt[2 (1/1)]) (((2^2)/(1 3))^(1/3)) ((((2^3) 4)/(1 (3^3)))^(1/4)) ((((2^4) (4^4))/((3^6) 5))^(1/5))

((((2/1)))^(1/2))(((((2^2)/(1 3))))^(1/3))((((((2^3)4)/(1 (3^3)))))^(1/4))((((((2^4)(4^4))/(1 (3^6) 5))))^(1/5))//N

Exp[EulerGamma]//N

e==((2 (1/1))^(1/1)) (Sqrt[((2^2)/(1 3))]) ((((2^3) 4)/(1 (3^3)))^(1/3)) ((((2^4) (4^4))/((3^6) 5))^(1/4))

((((2/1)))^(1/1))(((((2^2)/(1 3))))^(1/2))((((((2^3)4)/(1 (3^3)))))^(1/3))((((((2^4)(4^4))/(1 (3^6) 5))))^(1/4))//N

E//N

Limits

expr=(((n+1)^(n+1))/(n^n))-((n^n)/((n-1)^(n-1)));

Limit[expr,n->Infinity]

Nested Series

FoldList[1/#2(1+#1)&,a[4],Reverse@Array[a,3]]

ENested[n_]:=1+Fold[1/#2(1+#1)&,n+1,Range[n,1,-1]]

{#,N[#,10]}&/@ENested/@Range[10]//TableForm

Continued Fractions

Plot

Block[{$MaxExtraPrecision=1000},Show[PixelPerfect[BasePlot[ContinuedFraction[E,256]],BlockSize->2]]
]

Simple Continued Fraction

ContinuedFractionK[Piecewise[(({{((2 (n+1))/3), n mod 3==2}, {1, True}}))],{n,∞}]+2

ContinuedFraction[E,50]

Prepend[Table[Piecewise[{{2(n+1)/3,Mod[n,3]==2},{1,True}}],{n,49}],2]

N[FromContinuedFraction[Prepend[Table[Piecewise[{{2(n+1)/3,Mod[n,3]==2},{1,True}}],{n,1000}],2]],50]

ContinuedFractionK[Piecewise[{{2(n+1)/3,Mod[n,3]==2},{1,True}}],{n,∞}]//InputForm

N[E,50]

cf=ContinuedFraction[E,50];
conv=FromContinuedFraction/@Reverse[NestList[Most,cf,Length[cf]-1]]

<<Utilities`Plot`

ListPlot[conv,PlotStyle->{Red,PointSize[.02]}]

Need V6

CorrectDigits[{E,#},1000]&/@conv

ContinuedFractionK[Piecewise[{{1, n == 1}, {4 n - 2, True}}]==(E-1)/2

ContinuedFractionK[Piecewise[{{1,n==1},{4n-2,True}}],{n,∞}]//InputForm

N[ContinuedFractionK[Piecewise[{{1,n==1},{4n-2,True}}],{n,50}],20]

N[((E-1)/2),20]

FromContinuedFraction[{0,1,6,10,14,18}]//N

ContinuedFractionK[4 n-2,{n,∞}]==((e-1)/(e+1))

ContinuedFractionK[4n-2,{n,∞}]

N[ContinuedFractionK[4n-2,{n,50}],20]

N[((E-1)/(E+1)),20]

FromContinuedFraction[{0,2,6,10,14,18}]//N

Non-Simple

(1/(ContinuedFractionK[n,n,{n,∞}]))+1

1+1/ContinuedFractionK[n,n,{n,∞}]

Table[ContinuedFractionK[n,n,{n,k}],{k,10}]

1+1/Table[ContinuedFractionK[n,n,{n,k}],{k,10}]

ContinuedFractionK[n,n,{n,0,∞}]

2+1/Fold[#2+#2/#1&,a[6],Reverse@Array[a,5]]

ENonsimple[n_]:=2+1/Fold[#2+#2/#1&,n+1,Range[n,1,-1]]

ENonsimple/@Range[0,10]

Rational Power

ContinuedFraction[E^(1/2),50]

ContinuedFraction[E^(1/3),50]

ContinuedFraction[E^(1/4),50]

ContinuedFraction[E^(1/5),50]

ContinuedFraction[E^(2/3),50]

High-Water Marks

HighWaterMarks[ContinuedFraction[N[E,200]]]//Timing

Fractional parts

Figure

With[{h=.015,dy=.08,dx=.02,c=E},Graphics[
Table[With[{y=-n dy,frac=FractionalPart[c Range[n]]},{
Text[n,{-dx,y},{1,0}],
Text[N@Last@frac,{1+dx,y},{-1,0}],
AbsolutePointSize[6],Point[{#,y}&/@frac],
Line[{{0,y},{1,y}}],
Line[{{#,y+h},{#,y-h}}]&/@Subdivide[0,1,n]
}],{n,10}]]]

Counts of fractional parts in equidistributed bins

A194341

(rows=Table[BinCounts[FractionalPart[E Range[n]],{0,1,1/n}],{n,20}])//Column

Numbers of empty intervals

A036412

Count[#,0]&/@rows

Table[Count[BinCounts[FractionalPart[E Range[n]],{0,1,1/n}],0],{n,95}]

Values of n  for which no bins are left blank

A036413

Position[rows,{1..}]//Flatten

BFileSequence[36417]

Position[Table[Count[BinCounts[FractionalPart[E Range[n]],{0,1,1/n}],0],{n,1000}],0]//Flatten//Timing

See also NumberTheory/EquidistributedSequence.nb

<<MathWorld`IntegerSequences`

AbsoluteTiming[Sequence@@Timing[EquidistributedSubdivisions[E,5*^5]]]

AbsoluteTiming[Sequence@@Timing[EquidistributedSubdivisions[E,1*^6]]]

AbsoluteTiming[Sequence@@Timing[EquidistributedSubdivisions[E,1*^7]]]

Non-convergent denominator terms

terms=BFileSequence[036413]

denom=Denominator[Convergents[E,20]]

Intersection[denom,terms]

Complement[terms,denom]

Mnemonic

NonZero[n_]:=If[n==0,False,True]

SetAttributes[NonZero,Listable];

Mnemonic::bad = "Mnemonic disagrees with constant `1` at digit `2` and word \"`3`\"";

Mnemonic[n_,s0_String]:=Module[
	{
s=StringReplace[s0,{"'"->""}],
str,l,dif,baddig,p,words
},
str=StringToStream[s];
	l=Map[StringLength,words=ReadList[str,Word,WordSeparators->{" ",",",".","?",":","!","$","-"}] /. {"O"->"","`"->""}];
	l=l /.{10->0,11->{1,1},12->{1,2},13->{1,3},14->{1,4},15->{1,5}}//Flatten;
	(* Kludge, but treat `!' by chucking ! and ' and changing ` to "" *)
	dif=l-RealDigits[n,10,Length[l]][[1]];
	baddig=Select[dif,NonZero,1];
	If[baddig=={},Length[l],
	Message[Mnemonic::bad,n,p=Position[dif,baddig[[1]]][[1,1]],words[[p]]]
	]
]

ToNumber[n_,d_]:=Module[{r=RealDigits[n,10,d]},
StringInsert[StringJoin[ToString/@r[[1]]],".",1+r[[-1]]]
]

Mnemonic[E,"By omnibus I traveled to Brooklyn."]

Mnemonic[E,"To disrupt a playroom is commonly a practice of children."]

Mnemonic[E,"It enables a numskull to memorize a quantity of numerals."]

Mnemonic[E,"I'm forming a mnemonic to remember a function in analysis."]

Mnemonic[E,"He repeats: I shouldn't be tippling, I shouldn't be toppling here!"]

Mnemonic[E,"In showing a painting to probably a critical or venomous lady, anger dominates.  O take guard, or she raves and shouts."]

Mnemonic[E,"We present a mnemonic to memorize a constant so exciting that Euler exclaimed: `!'
when first it was found, yes, loudly `!'  My students perhaps will compute $e$, use power or Taylor series, an easy
summation formula, obvious, clear, elegant!"]

Khinchin and Khinchin-Lévy Constant

KhinchinList[x_,n_:200]:=MapIndexed[#1^(1/#2[[1]])&,Rest@FoldList[Times,1,Rest[ContinuedFraction[Mod[x,1],n+1]]]]
KhinchinLevyList[x_,n_:200]:=MapIndexed[#1^(1/#2[[1]])&,Denominator/@FromContinuedFraction/@Reverse[NestList[Drop[#,-1]&,Rest[ContinuedFraction[Mod[x,1],n+1]],n]]]

<<Utilities`Typesetting`

Block[{$MaxExtraPrecision=2000},With[{k=1000,c=E},
g1=ListPlot[KhinchinList[c,k],Joined->True,PlotStyle->Red,AxesLabel->TraditionalForm/@{n,((Subscript[a, 1])(Subscript[⋯a, n]))^InlineFraction[1,n]},
DisplayFunction->Identity,
PlotLabel->StyleForm["Khinchin's constant",FontFamily->"Times",FontSlant->"Italic"],
Epilog->{Blue,Dashing[{.02}],Line[{{0,#},{k,#}}]&[2.685452001]}];
g2=ListPlot[KhinchinLevyList[c,k],Joined->True,PlotStyle->Red,AxesLabel->TraditionalForm/@{n,(Subscript[q, n])^InlineFraction[1,n]},
DisplayFunction->Identity,
PlotLabel->StyleForm["Khinchin-Lévy constant",FontFamily->"Times",FontSlant->"Italic"],
Epilog->{Blue,Dashing[{.02}],Line[{{0,#},{k,#}}]&[Exp[Pi^2/12/Log[2]]]}
];
]]//Timing

Show[GraphicsArray[{g1,g2}]]

BBP-Type

Search

<<Utilities`Simplify`

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

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

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

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

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

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

Monitor[Do[If[(b=FindBasis[E,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,1}],{-1,n,p}];//Timing

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

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

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

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

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

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

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

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

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

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