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MathGroup Archive 1997

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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


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