Re: abundance of irrationals!)

"*** T. Winter" <***.Winter@xxxxxx> wrote in message news:<IELt4E.K2G@xxxxxx>...
> 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.

The standard meaning of Cauchy's convergence criterion is to find out
something for ALL n > n_0. This implies an infinite sum, but not the
limit n-->oo the existence of which you want to establish by using
Cauchy's criterion.

>
> > 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?

You oppose to infinite sums but you need a proof for ALL n > n_0 which
ultimately leads to an infinite sum, before you know whether the
series is converging.

Regards, WM
.

Loading