```Date: Oct 6, 2017 3:05 PM
Author: Markus Klyver
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.

Den fredag 6 oktober 2017 kl. 19:35:26 UTC+2 skrev John Gabriel:> On Friday, 6 October 2017 11:01:55 UTC-4, Markus Klyver  wrote:> > Den fredag 6 oktober 2017 kl. 16:16:51 UTC+2 skrev genm...@gmail.com:> > > On Thursday, 5 October 2017 19:55:38 UTC-4, Markus Klyver  wrote:> > > > Den onsdag 4 oktober 2017 kl. 22:38:55 UTC+2 skrev genm...@gmail.com:> > > > > On Wednesday, 4 October 2017 15:19:41 UTC-4, Markus Klyver  wrote:> > > > > > Den tisdag 3 oktober 2017 kl. 23:30:02 UTC+2 skrev John Gabriel:> > > > > > > On Tuesday, 3 October 2017 17:15:59 UTC-4, burs...@gmail.com  wrote:> > > > > > > > Can you make an example, where two series cannot> > > > > > > > be multiplied formally, independent of their> > > > > > > > convergence?> > > > > > > > > > > > > > Birdbrains.  It makes no sense to multiply any two series unless they both converge. The idea is that you are multiplying the LIMITs by multiplying the partial sums.> > > > > > > > > > > > > > YOU BIG MOOOOOOROOOOOOON!!!!> > > > > > > > > > > > You can, but it's no guarantee the product will converge or to what it will converge to. You can multiply two divergent series and get an convergent one, for example.> > > > > > > > > > Nonsense. Provide an example you ignoramus!> > > > > > > > Sure, consider two infinite sums with the general terms 1/n and 1/(n+1). Now, when multiplied, all terms will be at most 1/n^2. > > > > > > Prove it moron. That's just your assertion.> > > > > > > They will also be all positive, so by the squeeze theorem the product will converge.> > > > > > Bullshit.> > > > > > > > > > > Den onsdag 4 oktober 2017 kl. 22:40:03 UTC+2 skrev genm...@gmail.com:> > > > > On Wednesday, 4 October 2017 15:17:21 UTC-4, Markus Klyver  wrote:> > > > > > Den lĂ¶rdag 30 september 2017 kl. 07:30:40 UTC+2 skrev John Gabriel:> > > > > > > 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+...> > > > > > > > > > > > > > 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   > > > > > > > > > > > > > > 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> > > > > > > > > > > > When we talk about infinity, it's usually understood as a limit. Particularly, this applies to infinite sums which are limits of finite sums.> > > > > > > > > > I do not allow you to use concepts that are ill-formed or which I have not approved. Do you understand idiot?> > > > > > > > > > You don't know what you are talking about. I do.> > > > > > > > Then please show how these concepts are ill-formed. Just stating they are does not prove it.> > > > Do you agree that 1/n is positive? > > > Do you agree that 1/(n+1) is positive as well? Then you agree with 1/(n(n+1)) being positive as well. From n^2 ? (n(n+1)) it follows that 1/(n(n+1)) ? 1/n^2. So we have a sequence bounded from below by 0 and above by 1/n^2. > > Except that the product of the two series is NOT 1/(n(n+1)), you CRANK!Well, it will be bounded by that if you define multiplication in the naive manner; that is you try to multiply (a_1 + a_2 + a_3 + ...)(b_1 + b_2 + b_3 + ...) as usual. For absolutely convergent series, this will work and give you the limits of respective series multiplied. For example, multiplying the Taylor series of sin(x) in this fashion will give you sin^2(x).
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