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Path: utzoo!watmath!csc
From: csc@watmath.UUCP (Computer Sci Club)
Newsgroups: net.puzzle
Subject: Re: Balls in Bowl (no correct answers yet)
Message-ID: <6956@watmath.UUCP>
Date: Mon, 20-Feb-84 15:03:30 EST
Article-I.D.: watmath.6956
Posted: Mon Feb 20 15:03:30 1984
Date-Received: Tue, 21-Feb-84 07:47:26 EST
References: <210@pucc-i>
Organization: U of Waterloo, Ontario
Lines: 23

The problem is similar to finding the sum of the series

       100 - 1 + 100 - 1 + 100 - 1 ...

As written the sum of the above series is infinity (ie the limit
of the sequence of partial sums is infinity).  However the subseries
consisting of all negative terms is divergent.  Hence we cannot
rearange more than a finite number of terms and expect to have
a limit for the sequence of partial sums.  This is what is involved
in such arguments as "number the balls sequentialy and say that
machine B removes ball N at time noon - 1/N".
   If we assume the problem is find the sum of

       100 - 1 + 100 -1 .....

and use the standard definition of the sum of a series, then the
problem has a well defined answer, there will be an infinite number of
balls in the bowl at noon.  Else  the argument "machine B removes ball
N + K at time noon - 1/N" allows an arbitrary answer. (note also that
infinity = limit as e goes to zero of the number of balls in the
bowl at time noon - e)
                                      William Hughes