Re: Orlow cardinality question

In article <1118117722.007130.45740@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
poespam-trap@xxxxxxxxx says...
>
>
> Tony Orlow wrote:
> > In article <1118094238.005518.16810@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > poespam-trap@xxxxxxxxx says...
> > >
> > >
> > > Tony Orlow wrote:
> > > > In article <1118080381.774015.174770@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > > jirka@xxxxxx says...
> > > > > So all converging series are decreasing? How interesting! I would
> > > > > like to know where these "basic" rules are written. Obviously you are
> > > > > following some different "axiom system."
> > > > Please reference the following page from MathWorld concerning Divergence
> > > > Tests, and get back to me. I suppose if the values are negative, they
> > > > could be said to be increasing, but they are getting closer to 0, and
> > > > this isn;t the case with natural numbers anyway.
> > > >
> > > > http://mathworld.wolfram.com/DivergenceTests.html
> > >
> > > What you've missed is that lim(k->oo) u_k = 0 is not
> > > the same thing as "u_k is decreasing".
> > >
> > > If you need a further hint, consider the function
> > > sin(x)/x, which also goes to 0 as x->oo, but is
> > > clearly not "decreasing", not even in absolute
> > > value.
> > >
> > > If you need still a further hint, consider this:
> > > not all series are monotonic.
> > Okay, like I said above, if it's negative, it's increasing
>
> I figured you probably wouldn't understand my point, hence
> the hints. No, it's not necessarily "increasing and negative"
> either. Not all series are monotonic.
>
> > but tending toward zero,
> > and what you have offered is an oscillating function that
> > tends toward zero.
>
> Yes. The limit being zero has a precise meaning that
> does not require monotonicity, from either direction.
> It doesn't have to oscillate smoothly either. It could
> go to 10^100 at x=1000 and oscillate wildly.
Yeah, so I DID understand your point after all, didn't I? You might want
to read the entire post before responding. People here are so anxious to
declare each other wrong they don't take the time to read.
>
> > So, let me rephrase that, and thank you for your
> > correction. The infinite series cannot converge to a finite value unless
> > the values of the numbers in the series tend toward zero, that is, the
> > limit at n=oo is zero. Okay?
>
> Yes. "Tend toward zero" is too vague to be much use, but
> as I said, "limit is zero" has a very precise meaning.
Well does the limit of 1 approach 0? No. That was my point, weeks ago.
>
> > My point still stands, and the only reason
> > it was stated as specifically as it was, was that we were discussing the
> > set of positive, continulally increasing, natural numbers. Clear enough?
>
> I don't like the term "continually increasing" applied to
> things like numbers which have fixed values. So a little
> warning flag in my mind is saying that you're about to
> make an unwarranted leap based on this vague semantic
> jumble, as you have before. It's proof by vagueness.
No it's not. Each number doesn't have a limit. The overall range as the
set is constructed according to its inductive definition approaches a
limit which is infinite, or rather, has no limit, if you prefer.
>
> But you haven't made the leap yet, so carry on.
Thanks
>
> - Randy
>
>

--
Have a nice day

Tony
.

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