1,2

The denominators are given in A101029.

The limit s=lim(s(n),n->infty) with s(n) defined below equals 3*sum(Zeta(2*k+1)/2^(2*k),k=1..infty) with Euler's (or Riemann's) Zeta function. This limit is 3*(2*ln(2)-1)= 1.158883083...; see the Abramowitz-Stegun reference p. 259, eq. 6.3.15 with z=1/2 together with p. 258, eqs. 6.3.5 and 6.3.3.

This is the first member (m=2) of a family of rational partial sum sequences s(n,m)=(m-1)*m*(m+1)*sum(1/((m*k-1)*(m*k)*(m*k+1)),k=1..n) which have limit s(m)=lim(s(n,m),n->infty) = -(gamma + Psi(1/m)+m/2 + Pi*cot(Pi*x)/2), with the Euler-Mascheroni constant gamma and the digamma-function Psi. The same limit is reached by (m^2-1)*sum(Zeta(2*k+1)/m^(2*k),k=1..infty).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, pp. 258-259.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

W. Lang: Rationals s(n) and more.

a(n)=numerator(s(n)) with s(n)=6*sum(1/((2*k-1)*(2*k)*(2*k+1)), k=1..n).

s(3)= 6*(1/(1*2*3)+ 1/(3*4*5) + 1/(5*6*7)) = 79/70, hence a(3)=79 and A101029(3)=70.

Cf. A101627, A101629, A101631 members m=3, 4, 5.

Sequence in context: A026841 A026848 A026864 this_sequence A125348 A126506 A026897

Adjacent sequences: A101025 A101026 A101027 this_sequence A101029 A101030 A101031

nonn,frac,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 17 2004