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Numerator of partial sums of a certain series.
5

%I #20 Dec 01 2024 20:09:46

%S 1,47,6931,238657,4563655,526760263,45934377581,2852342564497,

%T 105651280880749,4335127472172929,186521117762900387,

%U 61393482232562091673,3255023127143379846869,3255958701070954680689

%N Numerator of partial sums of a certain series.

%C The denominators are given in A101630.

%C Third member (m=4) of a family defined in A101028.

%C The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 15*sum(Zeta(2*k+1)/4^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 15*(3*log(2)-2) = 1.1916231251...; see the Abramowitz-Stegun reference (given in A101028) p. 259, eq. 6.3.15 with z=1/4 together with p. 258.

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H W. Lang: <a href="/A101629/a101629.txt">Rationals s(n) and more.</a>

%F a(n)=numerator(s(n)) with s(n)=60*sum(1/((4*k-1)*(4*k)*(4*k+1)), k=1..n) = 15*sum(1/((4*k-1)*k*(4*k+1)), k=1..n).

%e s(3)= 60*(1/(3*4*5)+ 1/(7*8*9) + 1/(11*12*13)) = 6931/6006, hence

%e a(3)=6931 and A101630(3)=6006.

%Y Cf. A101028, A101627, A101631, members 2, 3, 5, resp.

%K nonn,frac,easy

%O 1,2

%A _Wolfdieter Lang_, Dec 23 2004