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From: "Joerg Buchholz" <buchholz@hs-bremen.de>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Radius of convergence?
Date: Tue, 10 Feb 2009 09:17:01 +0000 (UTC)
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"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message 
:
>   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

I just love this news group! You post a question, go to bed, have desperate dreams about limits and radii of convergence, wake up the next morning, and read such a highly sophisticated answer! Thank you Roger!

Some of my findings:

Seems like Mathematica has a function called 'SeriesCoefficient' that can return the nth coefficient as a symbolic function of n:

http://reference.wolfram.com/mathematica/ref/SeriesCoefficient.html

MuPAD has a similar function; it returns a symbolic sum for the nth coefficient:

series(exp(-x), x, infinity)

returns

sum(((-1)^k*x^k)/(k*gamma(k)), k = 0..infinity)

This seems to be a good starting point for the ratio test, because gamma(k)/gamma(k+1) = 1/k.
I am still learning how to use MuPAD syntax ...

MuPAD fails to find a symbolic sum for the expansion of tan(x); I cannot try that in Mathematica.

Wikipedia gives an example on how to find the radius of convergence if Bernoulli numbers are involved:

http://en.wikipedia.org/wiki/Radius_of_convergence#A_more_complicated_example