Date: Sep 30, 2017 4:14 AM
Author: Karl-Olav Nyberg
Subject: Re: The objects that Newton played with were called infinite series<br> but had ZERO to do with infinity. The name infinite series is a misnomer.

lørdag 30. september 2017 07.30.40 UTC+2 skrev John Gabriel følgende:
> The objects that Newton played with were called infinite series but had ZERO to do with infinity. The name infinite series is a misnomer.
>
> s = 1/2+1/4+1/8+... = 3/6+3/12+3/24+...
> t = 1/3+1/9+1/27+... = 2/6+2/18+2/54+...

Why are you expanding the fractions?
It's like you are preparing to add them.
>
> s * t = 6/36 + 6/108 + 3/108 + 3/108 + 1/12 + 3/324 + 1/24 +1/72 + 6/1296+...
>
> If my arithmetic is correct, then you end up getting:
>
> s * t = 6/36 + 12/108 + 24/324 + ... = 1/2

You know that you can not in general do this with series ?
KON
>
> So all Newton did was work with the LIMITS. Nothing with infinity. By taking sufficient terms he was able to calculate the product of the limits. So strictly speaking he is not multiplying series at all, ONLY some of the partial sums and from these obtaining the limit.
>
> Newton used this approach in determining sine series through inversion. He knew that he might end up with a series that could no longer be summed as in the case of these example geometric series, but he also knew that if he could find a pattern, then he would be able to approximate the sine ratio.
>
> This is hard evidence that it's a very bad idea to define S = Lim S.
>
> How arc length was derived:
>
> https://drive.google.com/open?id=0B-mOEooW03iLeGt2ZlViMzNyYTg
>
> No doubt the majority of the morons on this site will not be able to produce sufficient inference to reach an AHA moment. The orangutans will simply dismiss all of this without any serious study or consideration. Too bad.
>
> Comments are unwelcome and will be ignored.
>
> Posted on this newsgroup in the interests of public education and to eradicate ignorance and stupidity from mainstream mythmatics.
>
> gilstrang@gmail.com (MIT)
> huizenga@psu.edu (HARVARD)
> andersk@mit.edu (MIT)
> david.ullrich@math.okstate.edu (David Ullrich)
> djoyce@clarku.edu
> markcc@gmail.com