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.

1, 7, 11, 463, 4039, 35839, 320503, 575267, 25854247, 232557151, 298927153, 18830313487, 6778577311, 1525146340543, 13726182847159, 123535108753519, 1111813831298023, 2001263178349523, 90056808665990167, 810511140554958031

Offset

0,2

Comments

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.

The denominators are found in A096953.

log(2) = 2*Sum_{k>=1} ((1/3)^(2*k-1))/(2*k-1), from the Taylor series of log((1+x)/(1-x)) for x=1/3.

References

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

Links

Table of n, a(n) for n=0..19.

W. Lang, More comments.

Formula

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)).

Example

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

Crossrefs

Sequence in context: A201120 A164328 A301736 * A143602 A177999 A293220

Adjacent sequences: A096949 A096950 A096951 * A096953 A096954 A096955

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Jul 16 2004

Status

approved