Re: difficulty using FindRoot

*To*: mathgroup at smc.vnet.net*Subject*: [mg110290] Re: difficulty using FindRoot*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sat, 12 Jun 2010 05:31:00 -0400 (EDT)

Daniel Lichtblau wrote: > 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 Okay, I got that wrong. First, for negative q the terms actually alternate in pairs after the first. But that's not the real issue. The claim that they strictly decrease in magnitude is correct for |q|>=sqrt(2). For 1<|q|<sqrt(2) it is a different ballgame. Eventually they decrease and so you have convergence. But they can get arbitrarily large before that happens, hence it is not trivial to bound the values. I believe one can show the sum approaches -1 as q approaches 1 from the right, and that it decreases in magnitude as q grows. And I think something similar happens for q<-1. And these together would indicate that you cannot attain a value of pi-1 with real q. But I do not have proofs and as I mention above, the case where 1<|q|<sqrt(2) seems tricky. If I am seeing correctly, the unit circle is a natural boundary of convergence for the function f(1/q). That is to say, it is analytic inside the circle and does not have an analytic continuation past any point on the circle. Daniel Lichtblau Wolfram Research

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