A068566 Numerator of Sum_{k=1..n} 1/(k * 2^k).

1, 5, 2, 131, 661, 1327, 1163, 148969, 447047, 44711, 983705, 7869871, 102309709, 204620705, 31972079, 32739453941, 556571077357, 556571247527, 10574855234543, 42299423848079, 42299425233749, 84598851790183

Offset: 1

Author

Benoit Cloitre, Mar 25 2002

Keywords

• easy
• frac
• nonn

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.

The denominators of the second column are essentially A036289, A097064 and A134401. (End)

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.

Crossrefs

Cf. A068565.

Cf. A036289, A097064, A134401.

Programs

GAP

List([1..30], n-> NumeratorRat( Sum([1..n], k-> 1/(2^k*k)) ) ) # G. C. Greubel, Jun 30 2019

Magma

[Numerator( (&+[1/(2^k*k): k in [1..n]]) ): n in [1..30]]; // G. C. Greubel, Jun 30 2019

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 *)

PARI

vector(30, n, numerator(sum(k=1,n, 1/(k * 2^k)))) \\ Michel Marcus, Aug 07 2015

Sage

[numerator( sum(1/(2^k*k) for k in (1..n)) ) for n in (1..30)] # G. C. Greubel, Jun 30 2019

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)