OFFSET
1,2
COMMENTS
Sum_{k>=1} 1/(k * 2^k) = log(2).
From Paul Curtz, Jun 11 2019: (Start)
(Link) page 9:
T0 = 1/2 = 1/2
T1 = 1/2 + 1/8 = 5/8
T2 = 5/8 + 1/24 = 2/3
T3 = 2/3 + 1/64 = 131/192
T4 = 131/192 + 1/160 = 661/960
(T5 = 661/960 + 1/384 = 1327/1920)
... .
a(n)/A068565(n) is the first and the third column.
LINKS
Robert Israel, Table of n, a(n) for n = 1..1228
Paul Curtz, Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.
FORMULA
From Peter Bala, Feb 05 2024: (Start)
Integral_{x = 0..1} x^n/(1 + x)^(n+1) dx = log(2) - Sum_{k = 1..n} 1/(k * 2^k).
Hence a(n) = the numerator of Integral_{x = 0..1} ((1 + x)^n - x^n)/(1 + x)^(n+1) dx.
Integral_{x = 0..1/2} x^n/(1 - x) dx = Integral_{x >= 2} 1/(x^(n+2) - x^(n+1)) dx = log(2) - a(n)/A068565(n). (End)
MAPLE
map(numer, ListTools:-PartialSums([seq(1/k/2^k, k=1..100)])); # Robert Israel, Jul 10 2015
MATHEMATICA
Numerator[Accumulate[Table[1/(k 2^k), {k, 30}]]] (* Harvey P. Dale, May 11 2013 *)
a[n_]:=Log[2]-Hypergeometric2F1[1+n, 1+n, 2+n, -1]/(1+n);
Numerator[Table[Simplify[a[n]], {n, 1, 30}]] (* Gerry Martens, Aug 06 2015 *)
PROG
(PARI) vector(30, n, numerator(sum(k=1, n, 1/(k * 2^k)))) \\ Michel Marcus, Aug 07 2015
(Magma) [Numerator( (&+[1/(2^k*k): k in [1..n]]) ): n in [1..30]]; // G. C. Greubel, Jun 30 2019
(Sage) [numerator( sum(1/(2^k*k) for k in (1..n)) ) for n in (1..30)] # G. C. Greubel, Jun 30 2019
(GAP) List([1..30], n-> NumeratorRat( Sum([1..n], k-> 1/(2^k*k)) ) ) # G. C. Greubel, Jun 30 2019
CROSSREFS
KEYWORD
easy,frac,nonn
AUTHOR
Benoit Cloitre, Mar 25 2002
STATUS
approved