A262023 -id:A262023 - cp's OEIS frontend

A262031 Numerator of partial sums of a reordered alternating harmonic series.

1, 4, 5, 31, 247, 389, 1307, 15637, 13327, 187111, 199123, 353201, 6364777, 127056883, 23083451, 24191987, 579694957, 535076383, 13912332463, 43224283189, 40355946289, 1210479158981, 38689398709811, 72866186391697, 75054119011297, 77117026909777, 73105817107177, 2777117009412349

Offset: 0

Wolfdieter Lang, Sep 08 2015

For the denominators see A262022.
The reordered alternating harmonic series considered here is 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ... + ... - ...
The limit n -> infinity of the partial sums s(n) = a(n)/A262031(n) is 3*log(2)/2, approximately 1.03972077083991... For the decimal expansion see A262023.
Combining three consecutive terms of this series leads to the series b(0) + b(1) + ..., with b(k) = (1/2)*(8*k+5)/((4*k+1)*(4*k+3)*(k+1)). This produces partial sums 5/6, 13/140, 7/198, 29/1560, 37/3230, ..., which are given by s(3*n+2), n = 0, 1, .... Therefore, the limit is the same as the one given above, and it is obtained from Sum_{k=0..n} b(k) = (1/4)*Psi(n+5/4) + (1/4)*Psi(n+7/4) - (1/2)*Psi(n+2) + (3/2)*log(2), with the digamma function Psi(x).
This reordered alternating harmonic series appears as an example in the famous Dirichlet article, p. 319 (Werke I). Martin Ohm showed that for the reordering with alternating m consecutive positive terms followed by n negative terms (here n = 2 and m = 1) the sum becomes log(2) + (1/2)*log(m/n). See the reference, paragraph 8. p. 12-14. See also the Pringsheim reference.

			The first fractions s(n) (in lowest terms) are 1, 4/3, 5/6, 31/30, 247/210, 389/420, 1307/1260, 15637/13860, 13327/13860, 187111/180180, 199123/180180, 353201/360360, ...
The values s(10^n), for n=0..6, are (Maple 10 digits) [1.333333333, 1.105133755, 1.047114258, 1.040469694, 1.039795760, 1.039728271, 1.039721521], to be compared with 3*log(2)/2 (approximately 1.039720771).

P. G. Lejeune Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind, unendlich viele Primzahlen enthält, Abh. Preuss. Akad. Wiss. (1837) 45 -81; Werke I, 313-342.
M. Ohm, De nonnullis seriebus infinitis summandis, Berolini, 1839, Typis Trowitzschii et filli.
A. Pringsheim, Ueber die Werthveränderungen bedingt convergenter Reihen und Producte, Math. Ann. 22 (1838) 455-503.

Cf. A262022 (denominator), A262023, A058313, A058312, A002162.

Table[Numerator@ Sum[Which[Mod[k, 3] == 0, 3/(4 k + 3), Mod[k, 3] == 1, 3/(4 k + 5), True, -3/(2 (k + 1))], {k, 0, n} ], {n, 0, 27}] (* Michael De Vlieger, Jul 26 2016 *)

lista(nn) = {my(s = 0); for (k=0, nn, if (k%3==2, t = -3/(2*(k+1)), if (k%3==1, t = 3/(4*k+5), t = 3/(4*k+3))); s += t; print1(numerator(s), ", "););} \\ Michel Marcus, Sep 13 2015

a(n) = numerator(s(n)) with s(n) = Sum_{k=0..n} c(k), where c(k) = 3/(4*k+3), 3/(4*k+5), -3/(2*(k+1)) if k == 0, 1, 2 (mod 3), respectively.

A100046 Decimal expansion of -Pi/4 + (3*log(2))/2.

Original entry on oeis.org

2, 5, 4, 3, 2, 2, 6, 0, 7, 4, 4, 2, 4, 6, 9, 6, 5, 4, 5, 1, 0, 1, 8, 7, 3, 3, 6, 3, 6, 7, 3, 8, 9, 1, 3, 1, 0, 6, 3, 9, 5, 7, 8, 5, 1, 6, 9, 6, 6, 0, 6, 4, 2, 5, 9, 3, 7, 2, 8, 3, 8, 6, 6, 1, 6, 3, 1, 3, 6, 3, 3, 1, 3, 8, 2, 9, 8, 9, 8, 2, 3, 7, 5, 1, 7, 8, 6, 2, 8, 4, 1, 5, 9, 0, 9, 8, 7, 6, 4, 3, 1, 7

Offset: 0

Eric W. Weisstein, Oct 31 2004

			0.2543226074...

Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A003881, A014081, A262023.

RealDigits[(3*Log[2])/2-Pi/4,10,120][[1]] (* Harvey P. Dale, May 28 2018 *)

Equals Sum_{k>=1} A014081(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A262022 Denominator of partial sums of a reordered alternating harmonic series.

Original entry on oeis.org

1, 3, 6, 30, 210, 420, 1260, 13860, 13860, 180180, 180180, 360360, 6126120, 116396280, 23279256, 23279256, 535422888, 535422888, 13385572200, 40156716600, 40156716600, 1164544781400, 36100888223400, 72201776446800, 72201776446800

Offset: 0

Wolfdieter Lang, Sep 08 2015

See A262031 for this reordered alternating harmonic series with partial sums s(n).

			See A262031 for s(n), n=0..11, and s(10^n) for n=0..6.

Cf. A262031, A262023.

lista(nn) = {my(s = 0); for (k=0, nn, if (k%3==2, t = -3/(2*(k+1)), if (k%3==1, t = 3/(4*k+5), t = 3/(4*k+3))); s += t; print1(denominator(s), ", "););} \\ Michel Marcus, Sep 13 2015

a(n) = denominator(s(n)) with s(n) = Sum_{k=0..n} c(k), where c(k) = 3/(4*k+3), 3/(4*k+5), -3/(2*(k+1)) if k == 0, 1, 2 (mod 3), respectively.

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A262031 Numerator of partial sums of a reordered alternating harmonic series.

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A100046 Decimal expansion of -Pi/4 + (3*log(2))/2.

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A262022 Denominator of partial sums of a reordered alternating harmonic series.

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