Re: difficulty using FindRoot
- To: mathgroup at smc.vnet.net
- Subject: [mg110264] Re: difficulty using FindRoot
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Thu, 10 Jun 2010 08:11:21 -0400 (EDT)
- References: <201006091121.HAA12047@smc.vnet.net>
Roger Bagula wrote: > 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 > 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
- References:
- difficulty using FindRoot
- From: Roger Bagula <roger.bagula@gmail.com>
- difficulty using FindRoot