Re: abundance of irrationals!)

In article <fb701d3c.0504091321.29a53f7a@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote in message
> news:<ITSnetNOTcom#virgil-E15DEF.17025407042005@xxxxxxxxxxxxxxxxxxxxxxxx>...
>
> > > >>NO! Where did you get the stupid notion that the limit need be
> > > >>"contained" in a series any more than in a sequence?
> > > >
> > > >
> > > > It is a stupid idea, in fact. And I am glad to see you are not sharing
> > > > it. But some people, even mathematicians, do not see a difference
> > > > between the limit
> > > >
> > > > oo
> > > > SUM (1/2^k) = 1
> > > > k=1
> > > >
> > > > and
> > > >
> > > > SUM (1/2^k)
> > > > k e N
> > > >
> > > > though they admit that
> > > >
> > > > lim {n --> oo} 1/2^n = 0 while for every natural number
> > > >
> > > > 1/2^n > 0
> > > >
> > > > Ins't that funny?
>
>
> > If Mueckenheim actually thinks that SUM[ k=1..oo, 1/2^k ] and
> > SUM{ k in N, 1/2^k ] are diferent, I challenge him to give the
> > mathematical definitions of each from which he allegedly can deduce
> > thedifference.
> >
> > Absent such evidence, his claim fails. And his qualification to teach
> > mathematics fails also.
>
> It is given above:
> > >>NO! Where did you get the stupid notion that the limit need be
> > >>"contained" in a series any more than in a sequence?
> SUM[ k=1..oo, 1/2^k ] contains the limit, and
> SUM{ k in N, 1/2^k ] does not.

They are both series, and, in fact, the same series, but with slightly
different notation.

Do you consider SUM[ k=1..oo; 1/2^k ] and SUM[ k=1,2,...; 1/2^k ] to be
different? If so you delude yourself.
.

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