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A262031
Numerator of partial sums of a reordered alternating harmonic series.
3
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
0,2
COMMENTS
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.
FORMULA
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.
EXAMPLE
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).
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A262022 (denominator), A262023, A058313, A058312, A002162.
Sequence in context: A265708 A224219 A128867 * A326998 A270286 A271803
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Sep 08 2015
STATUS
approved