vdb@hou2g.UUCP (R.VANDERBEI) (01/10/85)
Lets get it straight! Real numbers are by definition limits of rationals:
x = lim r(n).
Hence any real number can be written as an infinite series of rational
numbers:
x = r(0) + [r(1)-r(0)] + [r(2)-r(1)] + ...
An infinite series is by definition a limit of partial sums. In fact the
notions are equivalent: every infinite series can be written as a limit
and every limit can be written as an infinite series.
.wjafyfe@watmath.UUCP (Andy Fyfe) (01/31/85)
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
until you exceed your desired limit, then negative ones until you're
below it, etc.
--andy fyfe ...!{decvax, allegra, ihnp4, et. al}!watmath!wjafyfe
wjafyfe@waterloo.csnetndiamond@watdaisy.UUCP (Norman Diamond) (01/31/85)
> 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 UUCP: {decvax|utzoo|ihnp4|allegra|clyde}!watmath!watdaisy!ndiamond CSNET: ndiamond%watdaisy@waterloo.csnet ARPA: ndiamond%watdaisy%waterloo.csnet@csnet-relay.arpa "Opinions are those of the keyboard, and do not reflect on me or higher-ups."
wjafyfe@watmath.UUCP (Andy Fyfe) (02/01/85)
In article <6909@watdaisy.UUCP> ndiamond@watdaisy.UUCP (Norman Diamond) writes: >Algorithms are supposed to execute in a finite length of time. Sigh. Semantics. The book "An introduction to the General Theory of Algorithms" (Machtey and Young) defines an algorithm as "a recipe or specific set of rules or directions for performing a task". Nothing about time. The word "finite" is particularly interesting, as the guarantee of termination was the condition we gave up (over the ability to enumerate programs) to get the effectively computable functions. (A bit of CS is creeping in here, but math-types don't worry too much about finiteness.) In any event, I thought the result was quite nice, and, at first, unexpected. Wonder what Norman would have thought had I merely given an existential "pseudo-proof-overview"! --andy fyfe ...!{decvax, allegra, ihnp4, et. al}!watmath!wjafyfe wjafyfe@waterloo.csnet