A294964 Numerators of the partial sums of the reciprocals of the positive numbers (k + 1)*(6*k + 5) = A049452(k+1).

1, 27, 1487, 71207, 423323, 5021921, 208393341, 19767960169, 9496615779853, 112702096556215, 7360072449683999, 524616965933727859, 526363371877036219, 43813027890740553917, 781806518388353706041, 148866078528885256002173, 15064339628673236669081953, 538212602352090865654383697

Offset: 0

Author

Wolfdieter Lang, Nov 27 2017

Keywords

• nonn
• frac
• easy

Comments

The corresponding denominators are given in A294965.

For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [6,5].

The limit of the series is V(6,5) = lim_{n -> oo} V(6,5;n) = . The value is (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) = 0.3135137477... given in A294966.

Examples

			The rationals V(6,5;n), n >= 0, begin: 1/5, 27/110, 1487/5610, 71207/258060, 423323/1496748, 5021921/17462060, 208393341/715944460, 19767960169/67298779240, 9496615779853/32101517697480, ...
V(6,5;10^6) = 0.313513577 (Maple, 10 digits) to be compared with the rounded ten digits 0.3135137478 obtained from V(6,5) given in A294966.
		

References

• Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.

Links

• Robert Israel, Table of n, a(n) for n = 0..640
• Eric Weisstein's World of Mathematics, Digamma Function

Crossrefs

Cf. A001620, A049452, A294512, A294966

Programs

Magma

[Numerator((&+[1/((k+1)*(6*k+5)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018

Maple

map(numer,  ListTools:-PartialSums([seq(1/(k+1)/(6*k+5),k=0..20)])); # Robert Israel, Nov 29 2017

Mathematica

Table[Numerator[Sum[1/((k+1)*(6*k+5)), {k,0,n}]], {n,0,25}] (* G. C. Greubel, Aug 29 2018 *)

PARI

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(6*k + 5)))); \\ Michel Marcus, Nov 27 2017

Formula

a(n) = numerator(V(6,5;n)) with V(6,5;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 5)) = Sum_{k=0..n} 1/A049452(k+1) = Sum_{k=0..n} (1/(k + 5/6) - 1/(k + 1)) = -Psi(5/6) + Psi(n+11/6) - (gamma + Psi(n+2)) with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.