Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: Notesfiles; site ea.UUCP Path: utzoo!watmath!clyde!floyd!harpo!ihnp4!inuxc!pur-ee!uiucdcs!ea!mwm From: mwm@ea.UUCP Newsgroups: net.puzzle Subject: Re: Balls in the bowl: Final word. - (nf) Message-ID: <10200001@ea.UUCP> Date: Sat, 3-Mar-84 07:25:00 EST Article-I.D.: ea.10200001 Posted: Sat Mar 3 07:25:00 1984 Date-Received: Wed, 11-Apr-84 07:38:06 EST References: <220@pucc-i.UUCP> Lines: 39 Nf-ID: #R:pucc-i:-22000:ea:10200001:000:1760 Nf-From: ea!mwm Mar 3 06:25:00 1984 #R:pucc-i:-22000:ea:10200001:000:1760 ea!mwm Mar 3 06:25:00 1984 /***** ea:net.puzzle / hou2g!stekas / 1:27 am Feb 27, 1984 */ I am still unconvinced that the problem of the balls in the bowl can have any other answer than infinity. At every time t=12:00-1/n there are 99n balls in the bowl. Therefore the number of balls is monotonically increasing without limit as noon is approached. One can play games with the terms in an infinite series only if it converges. Jim /* ---------- */ You are right, the answer is infinity, as in what you get when you divide any non-zero number by zero. In that case, as in this one, we write down the tripped-eight symbol to indicate that the answer is indeterminate. Key fact to note: infinity, per se, is not a number, and cannot be manipulated like one. The tripped-eight symbol we (meaing mathematicians) read as infinity actually has two seperate and distinct meanings, depending on context. When you confuse these contexts, you start getting answers where there aren't any. The first meaning is the one above (indeterminate). The second occurs only with limits and such, as the bound on a variable. This means (and should be read as) "increases without bound." In other words, goes to Aleph-null (which IS a number, and CAN be manipulated like one) and maybe beyond. BTW - having a series converge IS NOT sufficient information to allow you to (safely) play games with a series. The series must converge absolutely, otherwise you can make it sum to anything (as in this case). Sorry to be so long-winded, but you hit a tender spot.