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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




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