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Subject: Re: Radius of convergence?
Date: Tue, 10 Feb 2009 02:24:01 +0000 (UTC)
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"Joerg Buchholz" <buchholz@hs-bremen.de> wrote in message <gmq63p$ik3$1@fred.mathworks.com>...
> ......
> Roger, 
> thank you very much for the precise redefinition of my question. Seems like we agree that Matlab cannot 'express that coefficient as a symbolic function of n' with on-board means. Do you think that there is a chance to write an m-file that could do that; or do you believe one could mathematically prove that this is not possible at all? 
> Joerg

  Writing such a clever m-file would be a profoundly difficult thing to do, Joerg.  To take a comparatively elementary example, consider the expansion of tan(x) about x = 0.  According to one of my texts, the x^(2*n-1) term has a coefficient of

 2^(2*n)*(2^(2*n)-1)/(2*n)!*Bn

where Bn is the n-th Bernoulli number.  Unfortunately there is no known single expression for Bn in terms of n as far as I know.  It apparently has to be generated using an iterative procedure involving Eulerian numbers starting with n = 1.  Taking limits would appear to require information about Bn that would be difficult to put in a form that a general 'limit' function would know how to handle.  If one can't hand 'limit' a specific symbolic expression in n, what kind of input, encompassing the properties of Bn as n approaches infinity, could one provide?  As is known, the "radius" of convergence here is abs(x) < pi/2, but how would one deduce this from the behavior of Bn?

Roger Stafford