Re: Root of Divergent Series

On Sun, 18 May 2008, Jon G. wrote:

> |E| is the magnitude of vector E.
> all limits are as n approaches infinity
> ln e = 1
>
> The power series 1 + x + x^2 + x^3 + ... + x^n  diverges when
> x>=1 and has the root,
>
Huh?  1 + x + x^2 +..+ x^n
is not a power series and the notion of divergence doesn't apply.

1 + x + x^2 + x^3 + ...
	is a power series and converges	iff x in (-1,1)

The root of what?
	1 + x + x^2 +..+ x^n = 0
or
	1 + x + x^2 + x^3 + ...  = 0
?

(1 - x^(n+1))/(1 - x) = 0

x = any (n+1)-st root of 1 except 1
and has a real solution iff x = -1 and n is even.

or 1/(1 - x) = 0;  no solution

> x = lim ln[(n!(n+1)|E|^2 - n!e^2)/(n!e-(n+1))]
>
> where \
>
> |E|^2=(1/0!)^2 + (1/1!)^2 + (1/2!)^2 + (1/3!)^2 + ... + (1/n!)^2
>
Huh?  E is not a constant.  Do you mean E_n?

> Proof
>
> 1+lim{(1+2+3+...+n)ln[(n!(n+1)|E|^2 - n!e^2)/(n!e-(n+1))]=0
>
> lim ln[(n!(n+1)|E|^2 - n!e^2)/(n!e-(n+1))]=-lim 1/(1+2+3+...+n)=0
>
> lim [(n!(n+1)|E|^2 - n!e^2)/(n!e-(n+1))]=1
>
> lim 1/(n!e-(n+1)) = lim 1/(n!(n+1)|E|^2 - n!e^2)
>
> 0=0
>
> E.O.P.
>
E.O.P ?

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