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