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A096952 Numerators of upper bounds for Lagrange remainder in Taylor's expansion of log((1+x)/(1-x)) for x=1/3, multiplied by 6/5. 3

%I

%S 1,7,11,463,4039,35839,320503,575267,25854247,232557151,298927153,

%T 18830313487,6778577311,1525146340543,13726182847159,123535108753519,

%U 1111813831298023,2001263178349523,90056808665990167,810511140554958031

%N Numerators of upper bounds for Lagrange remainder in Taylor's expansion of log((1+x)/(1-x)) for x=1/3, multiplied by 6/5.

%C An upper bound for the Lagrange remainder in the expansion of log((1+x)/(1-x)) for x=1/3, i.e., for log(2), is R(2*n) = (1/2^(2*n+1) + 1/3^(2*n+1))/(2*n+1), n>=0.

%C The denominators are found in A096953.

%C log(2)= 2*sum(((1/3)^(2*k-1))/(2*k-1),k=1..infty), from the Taylor series of log((1+x)/(1-x)) for x=1/3.

%D M. Barner and F. Flohr, Analysis I, de Gruyter, 5. Auflage, 2000; p. 293.

%H W. Lang, <a href="http://www.itp.kit.edu/~wl/EISpub/A096952.text">More comments</a>.

%F a(n)=numerator(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)).

%e n=3: R(2*3)=(5/6)* a(3)/A096953(3) = (5/6)*463/326592 = 2315/1959552 = 0.001181.., therefore log(2)-2*sum(((1/3)^(2*k-1))/(2*k-1), k=1..3) < 0.001181.. In fact, the partial sum is 0.0001430654..

%K nonn,easy,frac

%O 0,2

%A _Wolfdieter Lang_, Jul 16 2004

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