Re: difficulty using FindRoot

*To*: mathgroup at smc.vnet.net*Subject*: [mg110350] Re: difficulty using FindRoot*From*: Roger Bagula <roger.bagula at gmail.com>*Date*: Tue, 15 Jun 2010 02:27:38 -0400 (EDT)*References*: <201006091121.HAA12047@smc.vnet.net> <huqkov$2ij$1@smc.vnet.net>

On Jun 10, 5:11 am, Daniel Lichtblau <d... at wolfram.com> wrote: > RogerBagulawrote: > > 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 -> 80= 0, > > 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.bag... at gmail.com > > Correct, it cannot be done with real q. Just work with the sum (so the > target is Pi-1). > > For -1<=q<=1 the sum does not converge because terms grow in size > (slightly different behavior at the endpoints, but same conclusion: > divergence). > > For q>1 the sum is alternating and terms strictly decrease in magnitude. > So it converges. But the first term is negative, so the result must be > negative. > > For q<-1 again it is alternating with terms strictly decreasing in > magnitude, hence convergent. This time the first term is between 0 and > 1/2, so the result of the sum is between 0 and 1/2. > > Conclusion: for real valued q, the sum cannot be Pi-1. > > Daniel Lichtblau > Wolfram Research Daniel Lichtblau Nice reasoning, thanks. Roger Bagula

**References**:**difficulty using FindRoot***From:*Roger Bagula <roger.bagula@gmail.com>