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Search: id:A039661
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| A039661 |
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Decimal expansion of exp(Pi). |
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+0
10
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| 2, 3, 1, 4, 0, 6, 9, 2, 6, 3, 2, 7, 7, 9, 2, 6, 9, 0, 0, 5, 7, 2, 9, 0, 8, 6, 3, 6, 7, 9, 4, 8, 5, 4, 7, 3, 8, 0, 2, 6, 6, 1, 0, 6, 2, 4, 2, 6, 0, 0, 2, 1, 1, 9, 9, 3, 4, 4, 5, 0, 4, 6, 4, 0, 9, 5, 2, 4, 3, 4, 2, 3, 5, 0, 6, 9, 0, 4, 5, 2, 7, 8, 3, 5, 1, 6, 9, 7, 1, 9, 9, 7, 0, 6, 7, 5, 4, 9, 2
(list; cons; graph; listen)
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OFFSET
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2,1
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COMMENT
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e^pi and pi^e (A059850) differ hardly by 3% in magnitude. The determination of the inequality sign between them dispenses with their actual evaluation, the result being immediate from the basic facts pi>e and ln(x+1)<x for positive x, whence setting x=(pi/e)-1 (>0) yields ln(pi)<pi/e, or pi^e < e^pi.
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REFERENCES
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L. Berggren, J. Borwein and P. Borwein, "Pi: a source Book", second edition, Springer, p. 422
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LINKS
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D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 37 (2000), 407-436. Reprinted from Bull. Amer. Math. Soc. 8 (Jul 1902), 437-479. See Problem 7.
S. Plouffe, exp(pi) to 5000 digits
S. Plouffe, exp(pi) to 5000 digits
Eric Weisstein. "Gelfond's Constant."
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FORMULA
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32*prod(j>=0, (u(j+1)/u(j))^2^(-j+1)) where u(0)=1 v(0)=1/sqrt(2) u(n+1)=u(n)/2+v(n)/2 v(n+1)=sqrt(u(n)v(n)) (deduced from Salamin algorithm for Pi) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 14 2003
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EXAMPLE
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23.1406926327792690...
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CROSSREFS
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Sequence in context: A122078 A126736 A127412 this_sequence A081877 A049076 A097744
Adjacent sequences: A039658 A039659 A039660 this_sequence A039662 A039663 A039664
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KEYWORD
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nonn,cons
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AUTHOR
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njas
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