A294965 -id:A294965 - cp's OEIS frontend

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

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

Offset: 0

Wolfdieter Lang, Nov 27 2017

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.

			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.

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

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

Cf. A001620, A049452, A294512, A294966

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

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

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

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

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.

A294966 Decimal expansion of the sum of the reciprocals of the numbers (k+1)(6k+5) = A049452(k+1) for k >= 0.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 27 2017

In the Koecher reference v_6(5) = (1/6)*(present value V(6,5)) = 0.05225229129512..., given on p. 192 as (1/4)*log(3) + (1/3)*log(2) - Pi/(4*sqrt(3)).

			0.313513747770728380036214711836908094696136733315523822488574116360843...

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

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Digamma Function.

Cf. A001620, A049452, A294964/A294965.

SetDefaultRealField(RealField(100)); R:= RealField(); (3/2)*Log(3) + 2*Log(2) - (1/2)*Pi(R)*Sqrt(3); // G. C. Greubel, Sep 05 2018

RealDigits[-PolyGamma[0, 5/6] + PolyGamma[0, 1], 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)

default(realprecision, 100); (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) \\ G. C. Greubel, Sep 05 2018

Sum_{k>=0} 1/((6*n + 5)*(n + 1)) =: V(6,5) = (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) = -Psi(5/6) + Psi(1) with the digamma function Psi and Psi(1) = -gamma = A001620.
The partial sums of this series are given in A294964/A294965.
Equals Sum_{k>=2} zeta(k)/6^(k-1). - Amiram Eldar, May 31 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A294966 Decimal expansion of the sum of the reciprocals of the numbers (k+1)(6k+5) = A049452(k+1) for k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A294966 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(6*k+5) = A049452(k+1) for k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

A294966 Decimal expansion of the sum of the reciprocals of the numbers (k+1)(6k+5) = A049452(k+1) for k >= 0.