Re: difficulty using FindRoot

*To*: mathgroup at smc.vnet.net*Subject*: [mg110237] Re: difficulty using FindRoot*From*: "David Park" <djmpark at comcast.net>*Date*: Thu, 10 Jun 2010 08:06:26 -0400 (EDT)*References*: <4839036.1276083624667.JavaMail.root@n11>

Hello Roger, It looks to me like there is no root in that region. The nearest approach I could find was: f[x_] := 1 + Sum[1/Product[1 - x^i, {i, 1, n}], {n, 1, 100}] Plot[f[x], {x, 1.0247, 1.0249}, PlotRange -> All, PlotRangePadding -> Scaled[.1], Frame -> True, Axes -> {True, False}, AxesOrigin -> {Automatic, 0}, WorkingPrecision -> 55] But this function is certainly difficult to explore. David Park djmpark at comcast.net http://home.comcast.net/~djmpark/ From: Roger Bagula [mailto:roger.bagula at gmail.com] The question of a q-form infinite exponential series solving to give Pi came up. I had absolutely no luck with infinite sums on this! I tried a plot of the function to narrow it down: Clear[f, x, n, i] f[x_] := 1 + Sum[1/Product[1 - x^i, {i, 1, n}], {n, 1, 100}] Plot[f[x], {x, 1.021831198825114750405873564886860549451, 1.02183648425181683450091441045515239239}, PlotRange -> All] The find root that seemed to work was: q /. FindRoot[1 + Sum[1/Product[1 - q^i, {i, 1, n}], {n, 1, 150}] - Pi == 0, {q, 1.0218701842518167}, WorkingPrecision -> 800, AccuracyGoal -> 795] gives: 1.0218311988251147504058736 with error messages: \!\(Divide::"infy" \(\(:\)\(\ \)\) "Infinite expression \!\(3.14159265346825122833252`25.0094071873645\/0\) \ encountered."\) \!\(\* RowBox[{\(FindRoot::"jsing"\), \(\(:\)\(\ \)\), "\<\"Encountered a singular Jacobian at the point \\!\\({q}\\) = \ \\!\\({1.0218311988251147504058736`25.0094071873645}\\). Try perturbing the \ initial point(s). \\!\\(\\*ButtonBox[\\\"More=85\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"FindRoot::jsing\\\"]\\)\"\>"}]\) 1 + Sum[1/Product[1 - (1.02183119882511475040587356488686054945)^i, {i, 1, n}], {n, 1, 150}] gives 0*10^(-19) It appears there is no real q such that the sum? 1 + Sum[1/Product[1 - q^i, {i, 1, n}], {n, 1, Infinity}]==Pi Respectfully, Roger L. Bagula 11759 Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : http://www.google.com/profiles/Roger.Bagula alternative email: roger.bagula at gmail.com