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Search: id:A096953
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| A096953 |
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Denominators of upper bounds for Lagrange-remainder in Taylor's expansion of ln((1+x)/(1-x)) multiplied by 6/5. |
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2
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| 1, 108, 1296, 326592, 15116544, 665127936, 28298170368, 235092492288, 47958868426752, 1929639176699904, 10968475320188928, 3027299188372144128, 4738381338321616896, 4605706660848611622912, 178087324219479649419264
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OFFSET
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0,2
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COMMENT
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An upper bound for the Lagrange-remainder in the expansion of ln((1+x)/(1-x)) for x=1/3, i.e. for ln(2), is R(2*n):=(1/2^(2*n+1) + 1/3^(2*n+1))/(2*n+1).
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REFERENCES
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M. Barner and F. Flohr, Analysis I, de Gruyter, 5te Auflage, 2000; p. 293.
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LINKS
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W. Lang, More comments.
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FORMULA
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a(n)=denominator(A(n)), where A(n):=(6/5)*(1/2^(2*n+1) + 1/3^(2*n+1))/(2*n+1) = A096951(n)/((2*n+1)*6^(2*n)).
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EXAMPLE
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n=4: R(2*4)=(5/6)* A096952(4)/a(4) = (5/6)*4039/15116544 = 20195/90699264 = 0.0002226589..., therefore ln(2)-2*sum(((1/3)^(2*k-1))/(2*k-1),k=1..4) < 0.0002226589... In fact, the partial sum is 0.0000124233...
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CROSSREFS
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Numerators are given in A096952.
Adjacent sequences: A096950 A096951 A096952 this_sequence A096954 A096955 A096956
Sequence in context: A059436 A129027 A101213 this_sequence A115135 A063809 A138784
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KEYWORD
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nonn,easy,frac
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jul 16 2004
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