```Date: Aug 18, 2006 11:46 AM
Author: Torsten Hennig
Subject: Re: Induction proof

>Hi Torsten, thank you for your reply however you're >solving a different>problem here. It appears that you've introduced a -1/n >to the right of>the inequality for your convinience. That isn't the >orignial problem.>I'm not sure what you're doing at all. Please everyone, >here's the>problem ((((( 1/2^2 + 1/3^2 + ... + 1/n^2 < 1 ))))) FOR >ALL n,>greater than or equal to 2, PROOF by INDUCTION. I only >capitalized for>clarity, not yelling here.>>Torsten Hennig wrote:>> >Prove by induction that 1/2^2 + 1/3^2 + ... + 1/n^2 < >1 >Please help!>> >Thank you!>>>> Hi,>>>> show by induction that>> 1/2^2 + 1/3^2 + ... + 1/n^2 < 1 - 1/n>> In the induction step, use that>> 1/(n+1)^2 < 1/(n*(n+1)) = 1/n - 1/(n+1) .>> >> Best wishes>> Torsten.Hi,you want to show that 1/2^2 + 1/3^2 + ... + 1/n^2 < 1 for all n >= 2.If you can show (e.g. by induction) that 1/2^2 + 1/3^2 + ... + 1/n^2 < 1 - 1/n for all n >= 2,you have what you want because 1 - 1/n < 1.Induction start : 1/2^2 = 1/4 < 1/2 = 1 - 1/2 ( < 1 )Induction step  :(1/2^2 + 1/3^2 + ... + 1/n^2) + 1/(n+1)^2 < (1 - 1/n) + 1/(n+1)^2   (by induction hypotheses) < (1 - 1/n) + 1/(n*(n+1))= (1 - 1/n) + (1/n - 1/(n+1))= 1 - 1/(n+1) ( < 1 )Best wishesTorsten.
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