[math-fun] A "numerical coincidence"
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
No, Maple just returns the sum, not evaluated. but.. my own inverter here found about 20 answers, and in the bunch : the exact expression : 1/2-1/4*ln(1+2^(1/2))*2^(1/2) I could have bet that Maple could do it. There is probably a way to force Maple to do it, with convert(), ? not certain. Also, another expression for that number here : 1/2-1/4*arctanh(1/2*2^(1/2))*2^(1/2) Also found using the inverter (the one with billion entries). Best regards, Simon Plouffe Le mer. 1 juil. 2020 à 07:12, Bill Gosper <billgosper@gmail.com> a écrit :
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
Oh for heaven's sake! Mike Hirschhorn points out that (c35) HYPERSIMP(SUM(X^N/BINOMIAL(2*N,N),N,0,INF)); Time= 20 msec. sqrt(x) asin(-------) sqrt(x) 2 4 (--------------------- + 1) x 2 sqrt(1 - -) 4 (d35) ----------------------------- (4-x) and In[779]:= Sum[x^n/Binomial[2 n, n], {n, 0,∞}] Out[779]= -((4 (2 Sqrt[4 - x] + Sqrt[x] ArcSin[Sqrt[x]/2]))/(Sqrt[4 - x] (-4 + x))) I.e., I'm losing it for forgetting hypersimp. And it's just a stupid bug in Mathematica: In[781]:= Sum[(-4)^n/Binomial[2 n, n], {n, 0, ∞}] // FunctionExpand Out[781]= Sum[(-4)^n/Binomial[2*n, n], {n, 0, Infinity}] —rwg On Tue, Jun 30, 2020 at 10:11 PM 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
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
participants (3)
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Bill Gosper -
James Buddenhagen -
Simon Plouffe