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A249403
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Decimal expansion of the maximum M(7) of the ratio (Sum_{k=1..7} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(7)) taken over x(1), ..., x(7) > 0.
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4
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1, 5, 8, 0, 0, 3, 7, 2, 1, 0, 6, 3, 2, 0, 5, 2, 3, 5, 2, 0, 8, 4, 0, 6, 3, 4, 9, 8, 1, 8, 3, 2, 6, 4, 4, 9, 2, 1, 1, 2, 8, 1, 5, 8, 0, 5, 9, 1, 6, 5, 9, 6, 1, 9, 7, 0, 1, 7, 4, 2, 3, 6, 9, 2, 0, 6, 0, 1, 5, 3, 7, 3, 7, 1, 0, 5, 3, 7, 7, 1, 1, 3, 5, 9, 2, 3, 5, 6, 4, 8, 0, 9, 0, 2, 1, 7, 0, 1, 4, 4, 8, 7, 0, 9, 0
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OFFSET
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1,2
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COMMENTS
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M(2) = (1+sqrt(2))/2, M(3) = 4/3.
M(n) = exp(1) - 2*Pi^2*exp(1)/(log(n))^2 + O(1/(log(n))^3), [de Bruijn, 1963].
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REFERENCES
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N. G. de Bruijn, Carleman's inequality for finite series, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag, Math., 25:505-514, 1963.
R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.
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LINKS
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EXAMPLE
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1.5800372106320523520840634981832644921128158059165961970174236920601537371...
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MATHEMATICA
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RealDigits[c7/.FindRoot[{1 + x2/2 + x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7, x2/2 + x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7*x2^2, x3/3 + x4/4 + x5/5 + x6/6 + x7/7 == c7*x3^3/x2^2, x4/4 + x5/5 + x6/6 + x7/7 == c7*x4^4/x3^3, x5/5 + x6/6 + x7/7 == c7*x5^5/x4^4, x6/6 + x7/7 == c7*x6^6/x5^5, x7/7 == c7*x7^7/x6^6}, {{c7, 3/2}, {x2, 1/2}, {x3, 1/2}, {x4, 1/2}, {x5, 1/2}, {x6, 1/2}, {x7, 1/2}}, WorkingPrecision->120], 10, 105][[1]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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