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Re: Analysis with series 1/2^2+1/3^2
Posted:
Aug 8, 2012 12:10 PM
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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.
>
>=
>
>(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 ?
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