difficulty using FindRoot

• To: mathgroup at smc.vnet.net
• Subject: [mg110231] difficulty using FindRoot
• From: Roger Bagula <roger.bagula at gmail.com>
• Date: Wed, 9 Jun 2010 07:21:29 -0400 (EDT)

```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\\\", \
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 :