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Re: Induction proof
Posted:
Aug 18, 2006 11:18 AM
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Paul Sperry, thank you for your reply, however you're attempting to
solve a different problem here. It seems that you've introducted the
number 1 to the problem as the first term in the series. I'm not sure
why you did this and again, the problem is proof by (induction).
Paul Sperry wrote:
> In article <1155835173.184893.228600@m79g2000cwm.googlegroups.com>,
> <emailtgs@gmail.com> wrote:
>
> > Prove by induction that 1/2^2 + 1/3^2 + ... + 1/n^2 < 1 Please help!
> > Thank you!
> >
>
> If S(n) = 1 + 1/2^2 + ... + 1/n^2 then lim(S(n), n -> oo) = Pi^2/6 =
> zeta(2). (Riemann's zeta function.) The proof(s) that zeta(2) = Pi^2/6
> are non-trivial and, as far as I know (which isn't _very_ far), not
> inductive.
>
> So, it _is_ true that 1/2^2 + ... + 1/n^2 < Pi^2/6 - 1 < 1. Proving
> that fact, independent of knowing zeta(2), will require some cleverness
> I think.
>
> --
> Paul Sperry
> Columbia, SC (USA)
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