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Topic: Analysis with series 1/2^2+1/3^2
Replies: 8   Last Post: Aug 9, 2012 12:17 PM

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dilettante

Posts: 141
Registered: 5/15/12
Re: Analysis with series 1/2^2+1/3^2
Posted: Aug 8, 2012 12:37 PM
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"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 ?

>




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