
Re: Induction proof
Posted:
Aug 18, 2006 12:21 AM


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 nontrivial 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)

