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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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Phil H

Posts: 49
Registered: 4/13/10
Re: Analysis with series 1/2^2+1/3^2
Posted: Aug 8, 2012 6:27 PM
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On Wednesday, August 8, 2012 7:23:02 AM UTC-7, Mina 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?) = (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 ?

Any series can be fluffed out to make it more challenging.
Unfluffed, this one is 1/(n^2+n)........1/2 + 1/6 + 1/12 + 1/20......
= 1/2 + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) ......
= 1
Phil H



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