On Wednesday, 5 March 2014 17:42:29 UTC+2, dull...@sprynet.com wrote: > On Tue, 4 Mar 2014 22:36:36 -0800 (PST), John Gabriel
> That's nonsense. The series that's just like the sine series except > that there's no x^7 term also converges. You cannot tell from the > first three terms whether the sine series is what it is or that > ogus one. Or one of infinitely many other possibilities...
I said you can tell from the general term and convergence. That is indeed how it's done and has been done.
> Btw you mentioned a point I didn't bother complaining about > since there were more basic problems. But as long as you > mention it: How do you know that there _is_ a convergent > power series for sin(x)?
There are many tests for convergence. Are you still a student? Once you complete your study, you'll know these things.
> It's in any decent calculus book.
Yet you don't know the answer, and still have not been able to provide any other explanation to these questions. :-)
> >> No, you never asked that. You made a sarcastic remark when I asked > >> how _you_ knew the general term. In any case: How one knows > >> the general term depends on how one defined sin(x).
Well, of course. This is so obvious that it hardly deserves mention. Is this really what was bothering you? Tsk, tsk.
> Starting from what you think is the definition, opposite over hypotenuse, > see any decent calculus book.
You are one very annoying man. I gave you an entire article that I wrote called How arc length was derived. Now either read it or shut up about this topic because you don't know enough to discuss it.
> Make up your mind! Are you claiming that R=(sqrt(2), oo)? Yesterday > you said R=[sqrt(2), oo).
Typo. It happens to all of us.
> Saying R=[sqrt(2), oo) as you did yesterday is simply wrong. > Saying R=(sqrt(2), oo) is not really right, but not as _simply_ > wrong: First problem is that (a,b) usually refers to all the > _reals_ between a and b; here R is the set of all _rationals_ > between sqrt(2) and infinity.
Right. sqrt(2) is that imaginary number between L and R. :-)
> Also it's totally missing the point. There's a _reason_ I gave > the definition of R the way I did instead of saying it was > the set of all rationals larger than sqrt(2). The reason is > that all this comes up when we're _constructing_ the reals;
There is no valid construction of the real numbers. The link I provided proves this irrefutably.
> at _that_ point we do not yet have any reason to bbelieve that there _is_ any > such thing as sqrt(2).
And still do not have any reason to think that sqrt(2) exists as anything else except an incommensurable magnitude.
> So saying R is the set of rationals larger than sqrt(2) > is getting things backwards.
Nonsense. It is exactly the set of rationals larger than sqrt(2) according to Dedekind. Of course there is no such set, so (L,R) is a misconstruct and hence invalid.
> We define x = (L,R), where L and R are as I said; _then_ it turns out > that x^2 = 2.
And if we define sqrt(cat) and then [sqrt(cat)]^2 = cat means it does not matter what you input for x to the function [sqrt(x)]^2 = x, because it will always be x. You don't really define anything, nor do you even get close to explaining what is sqrt(2).
> >> A real number is whatever the _definition_ of "real number" says > >> is a real number.
That's nonsense. A definition must be well formed. That's something most mainstreamers don't know or understand. You can't just make up definitions as you please - they must be well formed. :-)
> If we're using the Dedkind cut definition, then yes, (L,R) is a real number.
No, it's not a real number of any kind, because Dedekind's definition is junk!
> A large part of your problem is you don't seem to understand what a > _definition_ is.
Oh, I understand pretty well. It's you who is clueless. :-)
>But of course I am perfectly justified. :-) There is no number trapped between L and R.
> >> There's nothing there to refute! _Arguments_ are subject to refutation > >> - you don't give any arguments there, just assertions.
Anyone following the trail of comments will see through your lies and platitudes. :-) You can't fool them. But if course as long as you persist on that path, you won't learn anything, except perhaps to become more ignorant than you already are.