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Re: Analysis with series 1/2^2+1/3^2
Posted:
Aug 8, 2012 12:37 PM


"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message news:9p3528hcb1iaqmifjhlkcej9aij5mgagl6@4ax.com... > On Wed, 8 Aug 2012 07:23:02 0700 (PDT), Mina <mina_world@hanmail.net> > wrote: > >>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?) > > It's not just the associative law. But this manipulation > is ok. The reason it's ok is because you're dealing with > a sum of _positive_ terms; if you have a sum of positive > terms any sort of regrouping or reordering is ok.
Isn't it Fubini's theorem that justifies this, since we are not just rearranging the terms of a single series, but changing the order of a double summation?
> >> >>= >> >>(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 ? >



