e Approximations

Author

Eric W. Weisstein
January 25, 2009

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

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

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

Bamberger

Communicated 2024-03-13:

(1+.2^9^(6*7))^5^3^84

(1+((2/10))^9^(6*7))^5^3^84

with

.2^9^(6*7) == (1/5)^9^(6*7) == 1/(5^9^42) = 1/(5^3^84)

It should be correct to 8368428989068425943817590916445001887164 decimal places.

5^3^84Log[(1+((2/10))^9^(6*7))]

Series[Log[1+x],{x,0,2}]

5^3^84 (((2/10))^9^(6*7)-(1/2)(((((2/10))^9^(6*7)))^2))

5^3^84 ((2/10))^9^(6*7)(1-(1/2)((2/10))^9^(6*7))

Log[5^3^84 ((2/10))^9^(6*7)(1-(1/2)((2/10))^9^(6*7))]

FullSimplify[3^84Log[5]+9^(6*7)Log[(2/10)]]

Log[5^3^84 ((2/10))^9^(6*7)(-(1/2)((2/10))^9^(6*7))]

Log[-(1/2)5^3^84 (((((2/10))^9^(6*7)))^2)]

Log[-(1/2)]+3^84 Log[5]+2Log[(((2/10))^9^(6*7))]

Log[-(1/2)]+3^84 Log[5]+2 9^(6*7)Log[(2/10)]

N[%,20]

E^%

Barron

CorrectDigits[E,(Catalan^(EulerGamma-5/7))(Pi^(EulerGamma+2/7))]

N[{E,(Catalan^(EulerGamma-5/7))(Pi^(EulerGamma+2/7))},20]//Column

Castellanos

CorrectDigits[E,(2+(54^2+41^2)/80^2)]

CorrectDigits[E,(Pi^4+Pi^5)^(1/6)]

CorrectDigits[E,(271801/99990)]

CorrectDigits[E,(150-(87^3+12^5)/83^3)^(1/5)]

CorrectDigits[E,(4-(300^4-100^4-1291^2+9^2)/91^5)]

CorrectDigits[E,(1097-(55^5+311^3-11^3)/68^5)^(1/7)]

Davidson

Sept. 7, 2004

N[{E,163^(32/163)},20]//ColumnForm

CorrectDigits[E,163^(32/163)]

Take[Reverse[Sort[Flatten[Table[{Max[ContinuedFraction[n^(m/n) - E, 6]],
n, m}, {n, 1, 200}, {m, 1, n}], 1]]], 100]

Lafont

J. Lafont
May 16, 2008

N[{HarmonicNumber[8](1+(1/(80^2))),E},20]//Column

CorrectDigits[E,HarmonicNumber[8](1+(1/(80^2)))]

Pegg

Column[N[{E,3-(Sqrt[(5/63)])},20]]

CorrectDigits[E,3-(Sqrt[(5/63)])]

Sabey (pandigital)

Found by R.~Sabey in 2004 and posted on https://erich-friedman.github.io/mathmagic/0804.html until superseded by Bamberger's expression in March 2024

(((1+(9^(-(4^(7 6))))))^(3^(2^85)))

Take the logarithm once:

(3^(2^85))Log[(1+(9^(-(4^(7 6)))))]

Take the logarithm again:

(2^85)Log[3]+Log[Log[(1+(9^(-(4^(7 6)))))]]

Series[Log[1+x],{x,0,3}]

Series[Log[1-x],{x,0,3}]

(2^85)Log[3]+Log[(9^(-(4^(7 6))))(1-((9^(-(4^(7 6))))/2))]

(2^85)Log[3]+Log[(9^(-(4^(7 6))))]+Log[1-((9^(-(4^(7 6))))/2)]

(2^85)Log[3]- (4^(7 6))Log[(3^2)]+((9^(-(4^(7 6))))/2)

(2^85)Log[3]-2 (2^84)Log[3]+((9^(-(4^(7 6))))/2)

((9^(-(4^(7 6))))/2)

Take Log again:

-(4^(7 6))Log[9]-Log[2]

N[%]