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From: ndiamond@watdaisy.UUCP (Norman Diamond)
Newsgroups: net.math
Subject: Re: Any number as infinite series
Message-ID: <6909@watdaisy.UUCP>
Date: Thu, 31-Jan-85 12:39:56 EST
Article-I.D.: watdaisy.6909
Posted: Thu Jan 31 12:39:56 1985
Date-Received: Fri, 1-Feb-85 01:01:45 EST
References: <17957@lanl.UUCP> <28200048@uiucdcs.UUCP> <1088@aecom.UUCP> <385@hou2g.UUCP> <11209@watmath.UUCP>
Organization: U of Waterloo, Ontario
Lines: 28

> All you need is a series that converges, but is not absolutely
> convergent, say, for example,
> 	1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 ......
> Then, by re-arranging the terms, you can make this limit equal to
> anything you want.  The basic algorithm is to take positive terms
---interrupt---                 ^^^^^^^^^
> until you exceed your desired limit, then negative ones until you're
> below it, etc.
> --andy fyfe

Algorithms are supposed to execute in a finite length of time.

Let's consider a simplification of the problem:  we want to do just
one iteration, i.e. given a "starting point", we want to compute until
we have either exceeded the limit or dropped back below it, whichever
the case may be.  Now, this problem can be solved by an algorithm if
your limit is rational (and appropriately expressed), but not if your
limit is irrational.

If there's such a thing as (uncomputable)**2, that problem is one.

-- Norman Diamond

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