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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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