A294830 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+3) = A147874(k+2) for k >= 0.

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

Offset: 0

Author

Wolfdieter Lang, Nov 16 2017

Keywords

• nonn
• cons

Comments

In the Koecher reference v_5(3) = (2/5)*(present value V(5,3)) = 0.192680709798338..., given there as (1/4)*log(5) - (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 - 2*sqrt(5))/5).

Examples

			0.481701774495828777077075929361914755234187459374841804730459014188150...
		

References

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

Links

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

Crossrefs

Cf. A001620, A001622, A147874, A294828/A294829.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); phi:= (1 + Sqrt(5))/2; ((5/2)*Log(5) - (2*phi-1)*(Log(phi) + (Pi(R)/5)*Sqrt(7 - 4*phi)))/4; // G. C. Greubel, Aug 30 2018

Mathematica

RealDigits[((5/2)*Log[5] - (2*GoldenRatio - 1)*(Log[GoldenRatio] + (Pi/5)*Sqrt[7 - 4*GoldenRatio]))/4, 10, 100][[1]] (* G. C. Greubel, Aug 30 2018 *)

PARI

default(realprecision, 100); phi=(1+sqrt(5))/2; ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4 \\ G. C. Greubel, Aug 30 2018

Formula

Sum_{k>=0} 1/((5*n + 3)*(n + 1)) =: V(5,3) = ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4 = (1/2)*(-Psi(3/5) + Psi(1)) with the golden section phi =(1 + sqrt(5))/2 = A001622 with the digamma function Psi and Psi(1) = -gamma = A001620.

The partial sums of this series are given in A294828/A294829.