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A096953 Denominators of upper bounds for Lagrange-remainder in Taylor's expansion of log((1+x)/(1-x)) multiplied by 6/5. 1
1, 108, 1296, 326592, 15116544, 665127936, 28298170368, 235092492288, 47958868426752, 1929639176699904, 10968475320188928, 3027299188372144128, 4738381338321616896, 4605706660848611622912, 178087324219479649419264, 6853291511342734094893056 (list; graph; refs; listen; history; text; internal format)
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).

REFERENCES

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

LINKS

Table of n, a(n) for n=0..15.

W. Lang, More comments.

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)).

EXAMPLE

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...

PROG

(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

(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

CROSSREFS

Numerators are given in A096952.

Sequence in context: A202608 A202092 A202484 * A187302 A269277 A115135

Adjacent sequences:  A096950 A096951 A096952 * A096954 A096955 A096956

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jul 16 2004

STATUS

approved

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Last modified July 21 04:40 EDT 2019. Contains 325189 sequences. (Running on oeis4.)