```Date: Aug 19, 2006 1:08 PM
Author: mon
Subject: Re: Induction proof

I want to understand why1/(n+1)^2   < 1/(n*(n+1))it is because: 1/(n+1)^2  =  1/  [(n+1) * (n+1)] < 1 / [ n * (n+1)]or another wayn+1 > n(n+1)*(n+1)> n * (n+1)   (multiply both sides by n+1 which is positive)(n+1)^2       > n * (n+1)1 / (n+1)^2  < 1 / (n * (n+1)) (when we take reciprocal inequality changes direction)<emailtgs@gmail.com> wrote in message news:1156006369.232048.70870@i3g2000cwc.googlegroups.com...> 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.> Would you please explain. Thank you>>> Torsten Hennig wrote:>> >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 wishes>> Torsten.>
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