A096953 Denominators of upper bounds for Lagrange-remainder in Taylor's expansion of log((1+x)/(1-x)) multiplied by 6/5.

1, 108, 1296, 326592, 15116544, 665127936, 28298170368, 235092492288, 47958868426752, 1929639176699904, 10968475320188928, 3027299188372144128, 4738381338321616896, 4605706660848611622912, 178087324219479649419264, 6853291511342734094893056

Offset: 0

Author

Wolfdieter Lang, Jul 16 2004

Keywords

• nonn
• easy
• frac

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).

Examples

			n=4: R(2*4)=(5/6)* A096952(4)/a(4) = (5/6)*4039/15116544 = 20195/90699264 = 0.0002226589..., therefore log(2)-2*sum(((1/3)^(2*k-1))/(2*k-1),k=1..4) < 0.0002226589... In fact, the partial sum is 0.0000124233...
		

References

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

Links

• W. Lang, More comments.

Crossrefs

Numerators are given in A096952.

Programs

Magma

[Denominator((6/5)*(1/2^(2*n+1) + 1/3^(2*n+1))/(2*n+1)): n in [0..20]]; // Vincenzo Librandi, Jul 06 2015

PARI

vector(30, n, n--; denominator((6/5)*(1/2^(2*n+1) + 1/3^(2*n+1))/(2*n+1))) \\ Michel Marcus, Jul 06 2015

Formula

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)).