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

 Messages: [ Previous | Next ]
 dilettante Posts: 141 Registered: 5/15/12
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
> 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?
> =
>
> (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.

Date Subject Author
8/8/12 mina_world
8/8/12 dilettante
8/8/12 David C. Ullrich
8/8/12 dilettante
8/9/12 David C. Ullrich
8/8/12 Gottfried Helms
8/8/12 Phil H
8/9/12 Frederick Williams
8/9/12 David C. Ullrich