```Date: Aug 20, 2006 2:12 PM
Author: Brian M. Scott
Subject: Re: Induction proof

On 20 Aug 2006 10:29:37 -0700, <emailtgs@gmail.com> wrote in<news:1156094977.426428.218100@i42g2000cwa.googlegroups.com>in alt.math.undergrad:> okay, my question is, just how do you decide to go from here ... to> here.> < (1 - 1/n) + 1/(n+1)^2   (by induction hypotheses)> < (1 - 1/n) + 1/(n*(n+1))> I know you've made a note of what you did but I don't> understand how you can do that.The 'how' is trivial.  1/(n+1)^2 < 1/(n*(n+1)), so of course(1 - 1/n) + 1/(n+1)^2 < (1 - 1/n) + 1/(n*(n+1)): you're justadding 1 - 1/n to both sides.To see why 1/(n+1)^2 < 1/(n*(n+1)), if it isn't alreadyobvious, observe that (n+1)^2 > n*(n+1).  In even moredetail: n+1 > n, so multiplying both sides by n+1 yields(n+1)^2 > n*(n+1) if n+1 > 0, i.e., if n > -1.  Here youknow that n > 1, so certainly n > -1.The 'why' is so that you can split 1/(n*(n+1)) into partialfractions, one of which is 1/n that cancels the 1/n that youalready have.[...]Brian
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