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 just
adding 1 - 1/n to both sides.

To see why 1/(n+1)^2 < 1/(n*(n+1)), if it isn't already
obvious, observe that (n+1)^2 > n*(n+1). In even more
detail: 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 you
know that n > 1, so certainly n > -1.

The 'why' is so that you can split 1/(n*(n+1)) into partial
fractions, one of which is 1/n that cancels the 1/n that you
already have.

[...]

Brian