On Saturday, March 1, 2014 8:08:21 PM UTC+1, John Gabriel wrote: > On Saturday, 1 March 2014 19:20:50 UTC+2, dull...@sprynet.com wrote: > > > I asked how you find the derivative of sin(x). You showed me a proof > > > > > > > I asked how you find the derivative of sin(x). You showed me a proof > > using the power series. > > > > The power series was only used in the last couple of steps to show you that the numerator is indeed divisible by m+n. Using the power series DOES NOT contradict anything I have said. It does not invalidate anyone of my claims. > > > > > Now when I ask the obvious questions about that you say you didn't have to. > > > > I gave you valid reasons. Here is how arc length was derived without calculus: > > > > https://www.filesanywhere.com/fs/v.aspx?v=8b6c688d5b6375ac73a3 > > > > This means I am allowed to use power series. > > > > And if you think that I need the Taylor series, think again! Here is the Gabriel polynomial: > > > > https://www.filesanywhere.com/fs/v.aspx?v=8b6c688d5b63747d6b9b > > > > It is more rigorous than the Taylor series. > > > > >You should either answer my questions or _show_ me the proof without power series. > > > > As I said, I am allowed to use power series because it's not infinite. There is no such thing as an "infinite" series. There is no "infinite" ANYTHING! :-) > > > > > >> See, I thought the point was the New Calculus was going to avoid > > > >> limits. > > > > It does. :-) > > > > >How in the world do you make sense of a power series without limits. > > > > It's an approximation series which is identified by the fact of convergence. So, although it has a limit, the limit is not used, because in most cases, we don't know what it is! :-) > > > > > Not infinite except for being an infinite series, fine. > > > > The ellipsis, that is, ... does not make anything infinite. > > > > sin(x) =/= x - x^3/3! + x^5/5! ... > > > > sin(x) = LIMIT (x - x^3/3! + x^5/5! ... ) > > > > Big difference. > > > > > > > Also the proof that you have the fiirst three terms right. As well > > > as an expllanation of what you mean by that... > > > > We can tell by a only a few successive terms whether a series converges or not. We do not even need to know if it has a limit! :-) In most cases, that's how we show a series converges, that is, by investigating the given terms. I think a better than for sin(x) = x - x^3/3! + x^5/5! ... is "partial series" or "incomplete series". The word infinite is a misnomer. > > > > >Huh? Then show me the proof that just uses the first three terms. > > Also the proof that you have the fiirst three terms right. As well > > as an expllanation of what you mean by that... > > > > I am not sure what you mean here. Sine is an approximation in most cases. > > > > You can study the Gabriel Polynomial mentioned earlier. It contains ZERO limits. > > > > > What's the diifference? > > > > Huge. 1 is a number but sin(x) is not a number, in most cases it is an incommensurable magnitude that can only be approximated by a rational number.
I think there is some confusion here. JG doesn't use a Taylor series to prove his point, but a Taylor polynom. But of course we don't prove anything by using Taylor polynomes (We use it to approximate.). He thinks he actually can use polynomes to prove things. It's like giving examples in mathematics that this fits (aritmically), and then conclude that this is generally proved.