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Re: Analysis with series 1/2^2+1/3^2
Posted:
Aug 8, 2012 11:38 AM


"Mina" <mina_world@hanmail.net> wrote in message news:3f018aded42f4c24b30ebcb6b6ddcad8@googlegroups.com... > Hello teacher~ > > {(1/2^2) + (1/3^2) + (1/4^2) + ...} > + {(1/2^3) + (1/3^3) + (1/4^3) + ...} > + {(1/2^4) + (1/3^4) + (1/4^4) + ...} > + ... > >  > I have a solution.(ambiguous) > > Namely, > > {(1/2^2) + (1/3^2) + (1/4^2) + ...} > + {(1/2^3) + (1/3^3) + (1/4^3) + ...} > + {(1/2^4) + (1/3^4) + (1/4^4) + ...} > + ... > > = > > (1/2^2 + 1/2^3 + 1/2^4 + ...) > + (1/3^2 + 1/3^3 + 1/3^4 + ...) > + (1/4^2 + 1/4^3 + 1/4^4 + ...) > > (this associative law ? Really possible?)
I think this rearrangement is justified by Fubini's theorem. Have you covered that? By the way, what level of student is solving this problem? High School? (If so, wow!, and kudos) Undergraduate? Graduate?> > = > > (1/2^2)/{1(1/2)} > + (1/3^2)/{1(1/3)} > + (1/4^2)/{1(1/4)} > +... > > = (1/2) + (1/3).(1/2) + (1/4).(1/3) + ... > > = (1/2) + {(1/2)(1/3)} + {(1/3)(1/4)} + ... > > = 1 > >  > Hm... how do you think about it ?
Looks right to me.



