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Math Forum » Discussions » Math Topics » alt.math.undergrad.independent

Topic: Induction proof
Replies: 24   Last Post: Aug 22, 2006 4:11 PM

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Paul Sperry

Posts: 1,392
Registered: 12/6/04
Re: Induction proof
Posted: Aug 18, 2006 12:21 AM
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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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