Path: news.mathworks.com!not-for-mail 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) Organization: Hochschule Bremen Lines: 34 Message-ID: <gmrgmd$2dr$1@fred.mathworks.com> References: <gmp919$5sd$1@fred.mathworks.com> <gmpg2v$ba9$1@fred.mathworks.com> <gmpmo1$71h$1@fred.mathworks.com> <gmpvfe$jr3$1@fred.mathworks.com> <gmq63p$ik3$1@fred.mathworks.com> <gmqog1$s4i$1@fred.mathworks.com> Reply-To: "Joerg Buchholz" <buchholz@hs-bremen.de> NNTP-Posting-Host: webapp-03-blr.mathworks.com Content-Type: text/plain; charset="ISO-8859-1" Content-Transfer-Encoding: 8bit X-Trace: fred.mathworks.com 1234257421 2491 172.30.248.38 (10 Feb 2009 09:17:01 GMT) X-Complaints-To: news@mathworks.com NNTP-Posting-Date: Tue, 10 Feb 2009 09:17:01 +0000 (UTC) X-Newsreader: MATLAB Central Newsreader 11154 Xref: news.mathworks.com comp.soft-sys.matlab:517252 "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