Re: abundance of irrationals!)

In article <fb701d3c.0504070921.4dae99fd@xxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) writes:
> Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote in message news:<ITSnetNOTcom#virgil-90B626.13424406042005@xxxxxxxxxxxxxxxxxxxxxxxx>...
>
> >
> > SUM {1/2^k: k in N} is not defined, except as a limit, and that limit
> > is equal to 1.
>
> One cannot sum over ALL n?

Indeed. From the basic axioms you can only conclude things about finite
sums...

> How, the hell, can one find out whether the series is converging at
> all?

The above is not a series. You have first to *define* what the sum over
k in N is meaning. So long as you do not define it it is nothing. So
what do you *mean* with the sum over k in N? The standard meaning is:
lim{n -> oo} sum {k <= n} 1/2^k.

> For this sake one must prove for ALL n > n_0 that ALL the partial
> sums S_n satisfy
> 1 - S_n < eps.

Why do you think this is different from
sum {i = 1 ... oo} 1/2^k = 1
? You have a serious problem understanding limits. Or you still do not
comprehend that infinite sums are defined in terms of limits. Or you
have your own special meaning of the sum over the elements of a set.

> It is incredible! Cauchy's convergence criterion is wrong! Mathematics
> has been wrong for 200 years!

In what way is Cauchy's convergence criterion wrong?
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