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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 11:38 AM
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"Mina" <mina_world@hanmail.net> wrote in message
news:3f018ade-d42f-4c24-b30e-bcb6b6ddcad8@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.




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