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

```

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