Re: Series - how to now when they converge?

*To*: mathgroup at smc.vnet.net*Subject*: [mg9105] Re: Series - how to now when they converge?*From*: Daniel Lichtblau <danl>*Date*: Mon, 13 Oct 1997 23:33:19 -0400*Organization*: Wolfram Research, Inc.*Sender*: owner-wri-mathgroup at wolfram.com

Jarkko Valtteri Lempiinen wrote: > > Series[f,{z,z0,n}] seems to create a Laurent series of f with > Mathematica 3.0. How do I know, for which values of z does it converge? > And what if there are many concentric Laurent series? > > -- > Jarkko Lempi=3DE4inen Jarkko.Lempiainen at hut.fi > Studying at HUT phone: +358 9 468 2044 > www.hut.fi/~jlempiai +358 50 586 9 286 > Address: J=3DE4mer=3DE4ntaival 7A132, 02150 ESPOO, FINLAND Regard the result of Series as being a truncation of some corresponding infinite formal power series. When Series gives terms to negative integer degrees, there are only finitely many of them. Hence there is a pole at z0, and so the corresponding formal power series converges in a punctured disk around that point. The radius of convergence is the same as for that of the ordinary power series obtained by removing those finitely many terms that appear to negative powers. But this is not really relevant, because Series returns a truncated power series, not a formal series with infinitely many terms. The truncated series can be viewed as a (possibly Laurent) polynomial plus "small" terms. If you want to know in what neighborhood of z0 your approximation is "good" then you really need to know, by other means, how well it converges (if at all). One start would be to find the actual radius by locating the nearest singularity of your function. This is in general a difficult problem. Numeric and graphical (which are at heart numeric) means might be employed to get an idea for any particular function. Of course if you can compare your function asymptotically to a series of known behavior e.g. geometric, then basic methods (e.g. those taught in calculus) can provide good bounds of error. Daniel Lichtblau Wolfram Research danl at wolfram.com