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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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Gottfried Helms

Posts: 1,895
Registered: 12/6/04
Re: Analysis with series 1/2^2+1/3^2
Posted: Aug 8, 2012 12:24 PM
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Hello Mina -

Am 08.08.2012 16:23 schrieb Mina:
> 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,


I append the "zeta"-expression at each row:

>
> {(1/2^2) + (1/3^2) + (1/4^2) + ...} \\ = zeta(2) - 1
> + {(1/2^3) + (1/3^3) + (1/4^3) + ...} \\ = zeta(3) - 1
> + {(1/2^4) + (1/3^4) + (1/4^4) + ...} \\ = zeta(4) - 1
> + ...
>

because everything is convergent, you can reorder to get your sums
of geometric series.

It is also known, that
oo
Sum (zeta(k)-1) = 1
k=2

(zeta(k)-1) converges very fast to zero with increasing k.

Gottfried Helms



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