cp's OEIS Frontend

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

%I A262031 #43 Aug 11 2025 08:11:44
%S A262031 1,4,5,31,247,389,1307,15637,13327,187111,199123,353201,6364777,
%T A262031 127056883,23083451,24191987,579694957,535076383,13912332463,
%U A262031 43224283189,40355946289,1210479158981,38689398709811,72866186391697,75054119011297,77117026909777,73105817107177,2777117009412349
%N A262031 Numerator of partial sums of a reordered alternating harmonic series.
%C A262031 For the denominators see A262022.
%C A262031 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 + ... + ... - ...
%C A262031 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.
%C A262031 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).
%C A262031 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.
%H A262031 P. G. Lejeune Dirichlet, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k99435r/f332.image">Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind, unendlich viele Primzahlen enthält</a>, Abh. Preuss. Akad. Wiss. (1837) 45 -81; Werke I, 313-342.
%H A262031 M. Ohm, <a href="https://books.google.de/books?id=sO09AAAAcAAJ&amp;printsec=frontcover">De nonnullis seriebus infinitis summandis</a>, Berolini, 1839, Typis Trowitzschii et filli.
%H A262031 A. Pringsheim, <a href="https://gdz.sub.uni-goettingen.de/id/PPN235181684_0022">Ueber die Werthveränderungen bedingt convergenter Reihen und Producte</a>, Math. Ann. 22 (1838) 455-503.
%F A262031 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.
%e A262031 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, ...
%e A262031 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).
%t A262031 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 *)
%o A262031 (PARI) 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
%Y A262031 Cf. A262022 (denominator), A262023, A058313, A058312, A002162.
%K A262031 nonn,frac,easy
%O A262031 0,2
%A A262031 _Wolfdieter Lang_, Sep 08 2015