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 wishes

Torsten.