On Tue, 4 Mar 2014 22:36:36 -0800 (PST), John Gabriel <firstname.lastname@example.org> wrote:
>On Wednesday, 5 March 2014 01:29:26 UTC+2, dull...@sprynet.com wrote: > > >> Right. How do you know the general term? The fact that the >> first three terms are what they are simply doesn't say >> anything about the next term - saying "the first three >> terms" in reply to my question about how you know the >> general term is just silly. > >Of course you can tell the general term from the first three terms and the fact that the series converges.
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...
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)?
> It's high school math. I am surprised you don't know it. Still waiting for you to show me how you can do it otherwise... :-)
It's in any decent calculus book.
> >> Unless there's some rule in the New Calculus to the effect >> that any time you think you see a pattern that apparent >> pattern must be valid. In which case the New Calculus >> is simply wrong. > >You should not make irrelevant comments. That has nothing to do with the New Calculus. It's a well-known part of mainstream math. Did you learn any mainstream math? > >> 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). Starting >> from what you think is the definition, opposite over hypotenuse, >> see any decent calculus book. > >That is a typical strawman argument. It won't buy you any credits. I am still waiting for you to show me another method... > >> It's a long story (if all the details are included). > >Bullshit. > >> How I know the general term is irrelevant to the question of how >> a person finds the derivative of sin in the New Calculus. The >> method you showed me depends on knowning the general >> term of the power series. But you've given no clue how you >> know the power series, other than the first three terms. > >I gave you an entire article. You ought to try reading it. :-) How arc length was derived is the name of the article and the link is in one of my previous comments. > >> >Bullshit. L=(-oo,sqrt(2)) and R=[sqrt(2), oo) => (L,R) is a Dedekind cut. >> No it's not. This is why I used the word "ignorant" the other day. >> You feel free to say that Dedekind cuts are all wrong somehow, >> even though you simply don't know what a Dedekind cut _is_. > >Of course it is. sqrt(2) defined as a D. Cut is EXACTLY L=(-oo,sqrt(2)) and R=(sqrt(2), oo).
Make up your mind! Are you claiming that R=(sqrt(2), oo)? Yesterday you said R=[sqrt(2), oo).
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.
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; at _that_ point we do not yet have any reason to bbelieve that there _is_ any such thing as sqrt(2).
So saying R is the set of rationals larger than sqrt(2) is getting things backwards. We define x = (L,R), where L and R are as I said; _then_ it turns out that x^2 = 2.
> >You don't know what is a D. cut. I suggest further study! > >> A real number is whatever the _definition_ of "real number" says >> is a real number. If we're using the Dedkind cut definition, then >> yes, (L,R) is a real number. > >Nonsense. You don't get to make up ill-formed definitions as you please.
A large part of your problem is you don't seem to understand what a _definition_ is.
> > L=(-oo,sqrt(2)) and R=[sqrt(2), oo) > >is NOT a number of ANY kind. It is a UNION of two sets that says absolutely NOTHING about sqrt(2). Nice try! :-) > >> >But of course I am perfectly justified. :-) There is no number trapped between L and R. > >Don't know where you got that notion from. But you are correct without even knowing it!!! There is no number trapped between L and R because sqrt(2) DOES NOT exist as a number. :-) > >> >http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409 > >> There's nothing there to refute! _Arguments_ are subject to refutation >> - you don't give any arguments there, just assertions. > >Well, this will be my last response to you. I don't have time to waste on trolls.