Curiously, if you replace (-4) by any of (-3), (-2), (-1) maple (at least maple 2016) does compute a sum, giving, with (-3) for example: -(4/49)*21^(1/2)*arcsinh((1/2)*3^(1/2))+4/7 which using convert(%,ln) then becomes: -(4/49)*21^(1/2)*ln((1/2)*3^(1/2)+(1/2)*7^(1/2))+4/7 With (-4) maple just returns the stated sum, as Simon mentioned. The desired series (with (-4)), however, starts out: 1, -2, 8/3, -16/5, 128/35, -256/63, 1024/231, ... And the partial sums: 1, -1, 5/3, -23/15, 223/105, -611/315, 8639/3465, ... alternate in sign, so technically this series does not converge. To get convergence, one must parenthesize in pairs. Interestingly, maple does give an explicit answer for the partial sums: sum((-4)^n/binomial(2*n, n),n=0..N); in terms of the MeijerG function, which I had not heard about until today, namely, that partial sum is: 1/2+(-1)^N*MeijerG([[1], [3/2+N]], [[1, 2+N], []], 1)*Pi^(1/2)-(1/4)*2^(1/2)*ln(1+2^(1/2)). See: https://en.wikipedia.org/wiki/Meijer_G-function On Wed, Jul 1, 2020 at 12:12 AM Bill Gosper <billgosper@gmail.com> wrote:
ries: Sum[(-4)^n/Binomial[2*n, n], {n, 0, Infinity}] == 1/2 + Log[-1 + Sqrt[2]]/(2*Sqrt[2])
Can Maple do this?
Craziness: Plouffe's old Inverter turns it into a much easier arcsinh sum, that Mathematica can do. But Maple seemingly could't when Plouffe tabulated it. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun