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Bessel and Legendre's Equations - Finding Lower Bound

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The problem is to determine a lower bound for the radius of convergence for the two following equations. I am able to get p(x) and q(x) for both equations, but I'm confused on how to proceed. I would like to see the problem worked out and what the lower bound is.

Bessel's Equation:

Centered at 1

what I believe are p(x) and q(x)

p(x) = and q(x) =

Legendre's Equation

Centered at 0, and also try it centered at 1

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Solution Summary

The expert finds the lower bounds for Bessel and Legendre's equations.

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Please see the attached doc file and please do not hesitate to contact me through the BrainMass staff if there is any vagueness. Good Luck.

We have:

and we want the lower bound for the radius of convergence about x=1. First we divide the equation by x^2:

now the radius is at least equal to the minimum between the radii of 1/x and (x^2-v)/x^2 about x=1:

1/x=1/(1+(x-1))=1-(x-1)+(x-1)^2-(x-1)^3+... provided |x-1|<1 therefore the radius of convergence about x=1 for the first one is 1.

(x^2-v)/x^2=1-v/x^2
we can ...

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