Date: Oct 6, 2017 4:03 PM
Author: Markus Klyver
Subject: Re: There are no axioms or postulates in Greek mathematics, only in mythmatics.

Den onsdag 27 september 2017 kl. 00:27:30 UTC+2 skrev John Gabriel:
> On Tuesday, 26 September 2017 16:21:14 UTC-4, Markus Klyver wrote:
> > Den tisdag 26 september 2017 kl. 16:29:43 UTC+2 skrev John Gabriel:
> > > On Tuesday, 26 September 2017 08:25:00 UTC-4, Me wrote:
> > > > On Tuesday, September 26, 2017 at 1:22:55 PM UTC+2, John Gabriel wrote:
> > > > > On Tuesday, 26 September 2017 05:54:23 UTC-4, Me wrote:
> > > > > >
> > > > > > See his "Simple proof using basic number theory, that pi cannot be expressed
> > > > > > as a ratio of two integers", for example:
> > > > > > https://drive.google.com/file/d/0B-mOEooW03iLdkZ0NGlaSkpDc00/view
> > > > > >

> > > > > It's as solid as can be.
> > > >
> > > > Well, at least you have a sense of humor! :-)

> > >
> > > With you claiming that 1+1/2+1/3+... *IS* a limit, I need to have a sense of humour. Chuckle.

> >
> > If the limit would exist, then yes.

>
> That is not what you said idiot. And no, S = Lim S is not a good definition,
>

> > But the harmonic series diverges, so under that definition it makes no sense to assign the series a numerical value.
>
> It is a very stupid idea to assign numeric values to series that can't be summed. That is the core of Euler's Blunder and why I corrected him.


To "sum" an infinite series means to consider the limit of its partial sums.


Den onsdag 27 september 2017 kl. 13:07:35 UTC+2 skrev John Gabriel:
> On Tuesday, 26 September 2017 18:53:14 UTC-4, Me wrote:
> > On Wednesday, September 27, 2017 at 12:27:30 AM UTC+2, John Gabriel wrote:
> >

> > > And no, S = Lim S is not a good definition,
> >
> > Good to know, since you are *the only one* using this idiotic expression.
> >
> > On the other hand, I thought it has been endorsed by your fuck buddy WM?

>
> This is what you wrote moron!
>
> Lim S = 1/(1 + a) = S = 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... is defined to be Lim S = lim_{n to infinity} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n.
>
> All I did was show you how it is equivalent to S = Lim S.
>
> Klyver has already moved past the denial stage onto the next denial stage: refusing to admit that it is ill formed.
>
> Persist. You might get there before you die.


"S = Lim S" is not even ill-formed, because it doesn't mean anything. If you mean we define the series as the limit of the series, then you are just wrong. We don't.

Den onsdag 27 september 2017 kl. 14:20:13 UTC+2 skrev John Gabriel:
> On Monday, 25 September 2017 19:22:51 UTC-4, John Gabriel wrote:
> > https://www.linkedin.com/pulse/part-1-axioms-postulates-mathematics-john-gabriel
> >
> > https://www.linkedin.com/pulse/part-2-axioms-postulates-mathematics-john-gabriel
> >
> > https://www.linkedin.com/pulse/part-3-axioms-postulates-mathematics-john-gabriel
> >
> > https://www.linkedin.com/pulse/part-4-axioms-postulates-mathematics-john-gabriel
> >
> > https://www.linkedin.com/pulse/part-5-axioms-postulates-mathematics-john-gabriel
> >
> > 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

>
> Here is a quote from the only mainstream academic I respect on sci.math:
>
>
> Euler's teacher was Johann Bernoulli, the more conceited and less genial of the Bernoulli brothers (of course being "less genial" than Jakob B does not mean a reproach). Euler was even more genial than both and many others. Nevetheless here he applied the wrong concept.
>
> John Gabriel is completely correct when he says:
>
> 1. S = Lim S, is wrong
> 2. The series is not the limit.
> 3. 1/3 cannot be expressed in base 10 because 3 is not a prime factor of 10.
>
> Unfortunately the contrary belief has lead to the mess of transfinite set theory.
>
> Regards, WM



The series is DEFINED as the limit of its partial sums.

Den onsdag 27 september 2017 kl. 18:05:08 UTC+2 skrev John Gabriel:
> On Wednesday, 27 September 2017 11:55:44 UTC-4, konyberg wrote:
> > onsdag 27. september 2017 14.15.55 UTC+2 skrev John Gabriel følgende:
> > > On Wednesday, 27 September 2017 07:12:25 UTC-4, burs...@gmail.com wrote:
> > > > Am Mittwoch, 27. September 2017 13:07:35 UTC+2 schrieb John Gabriel:
> > > > > Lim S = 1/(1 + a) = S = 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... is
> > > >
> > > > Nope:

> > >
> > > YES MORON.
> > >

> > > > [S] = 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ...
> > > >
> > > > Is a shorthand for:
> > > > [Lim S] = lim_{n to infinity} (1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n).
> > > >
> > > > Hence:
> > > > [Lim S] = 1/(1+a) = 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ...
> > > >

> > >
> > > Hence S = Lim S according to Euler.
> > >
> > > Euler Oagbar! Euler Oagbar! Euler Oagbar! Euler is one! Euler is great! Euler Oagbar! Euler Oagbar! Euler Oagbar!
> > >
> > > Chuckle. You infinitely stupid baboon!

> >
> > This is wrong. Correct is:
> > S = lim(n to inf)Sum((-1^n)a^n). So S = lim S(n) (shorthand), not S = lim S. By your own math!
> > Euler never wrote S = lim S.
> > KON

>
> Nope. You are wrong.
>
> S IS the series. Why? Because it IDENTIFIES the mathematical object.
> LIM S is the limit and all it does is identify ONE attribute of the series. But many series can have the same limit. So Lim S is insufficient to identify the series. Thus, it makes perfect sense to identify the series by the identifier S. After all, you can't have 0.333... or 3.14159... if you don't have a "unique" series, no? Of course adding an ellipsis does not make it unique because there is no such thing as an infinite series, only in your syphilitic brain.
>
> Therefore, S = Lim S exactly as I explained. Euler most definitely wrote S = Lim S.
>
>
> I am fiercely intelligent and can see these things which you will never think of in all your life. You are privileged that I am bothering to respond to you - an absolute moron that you are.


Many finite sums are equal to each other as well, so what's your problem with series? A series is the limit of its partial sums. No one said all series had to have a different value.

Den onsdag 27 september 2017 kl. 18:18:02 UTC+2 skrev John Gabriel:
> On Wednesday, 27 September 2017 12:05:46 UTC-4, konyberg wrote:
> > onsdag 27. september 2017 14.20.13 UTC+2 skrev John Gabriel følgende:
> > > On Monday, 25 September 2017 19:22:51 UTC-4, John Gabriel wrote:
> > > > https://www.linkedin.com/pulse/part-1-axioms-postulates-mathematics-john-gabriel
> > > >
> > > > https://www.linkedin.com/pulse/part-2-axioms-postulates-mathematics-john-gabriel
> > > >
> > > > https://www.linkedin.com/pulse/part-3-axioms-postulates-mathematics-john-gabriel
> > > >
> > > > https://www.linkedin.com/pulse/part-4-axioms-postulates-mathematics-john-gabriel
> > > >
> > > > https://www.linkedin.com/pulse/part-5-axioms-postulates-mathematics-john-gabriel
> > > >
> > > > 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

> > >
> > > Here is a quote from the only mainstream academic I respect on sci.math:
> > >
> > >
> > > Euler's teacher was Johann Bernoulli, the more conceited and less genial of the Bernoulli brothers (of course being "less genial" than Jakob B does not mean a reproach). Euler was even more genial than both and many others. Nevetheless here he applied the wrong concept.
> > >
> > > John Gabriel is completely correct when he says:
> > >
> > > 1. S = Lim S, is wrong

> > Of course it is wrong! Euler never wrote that!
>
> Moron. He did write that.
>

> > > 2. The series is not the limit.
> > Of course not if a series is defined as finite.
>
> The series in discussion is called an infinite series you idiot. That statement is hand waving bullshit.
>

> > But if the series is defined as the infinite sum, then it is the limit.
>
> Wrong. That would be like saying 0.333... the UNIQUE representation of the series is the limit, but point 3 below shows this is not possible.
>
>

> > > 3. 1/3 cannot be expressed in base 10 because 3 is not a prime factor of 10.
> > This, and both 1. and 2. is your own invention. Who is the good professor It is WM!
>
> Oh, so you can express 1/3 in base 10? What a moron!
>
>

> >
> > KON

> > >
> > > Unfortunately the contrary belief has lead to the mess of transfinite set theory.
> > >
> > > Regards, WM



A series IS defined to be the limit of its partial sums though. 1/3 has an unique decimal expansion in base 10, namely 0.(3).

Den onsdag 27 september 2017 kl. 22:37:38 UTC+2 skrev John Gabriel:
> On Wednesday, 27 September 2017 12:41:50 UTC-4, konyberg wrote:
> > onsdag 27. september 2017 18.18.02 UTC+2 skrev John Gabriel følgende:
> > > On Wednesday, 27 September 2017 12:05:46 UTC-4, konyberg wrote:
> > > > onsdag 27. september 2017 14.20.13 UTC+2 skrev John Gabriel følgende:
> > > > > On Monday, 25 September 2017 19:22:51 UTC-4, John Gabriel wrote:
> > > > > > https://www.linkedin.com/pulse/part-1-axioms-postulates-mathematics-john-gabriel
> > > > > >
> > > > > > https://www.linkedin.com/pulse/part-2-axioms-postulates-mathematics-john-gabriel
> > > > > >
> > > > > > https://www.linkedin.com/pulse/part-3-axioms-postulates-mathematics-john-gabriel
> > > > > >
> > > > > > https://www.linkedin.com/pulse/part-4-axioms-postulates-mathematics-john-gabriel
> > > > > >
> > > > > > https://www.linkedin.com/pulse/part-5-axioms-postulates-mathematics-john-gabriel
> > > > > >
> > > > > > 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

> > > > >
> > > > > Here is a quote from the only mainstream academic I respect on sci.math:
> > > > >
> > > > >
> > > > > Euler's teacher was Johann Bernoulli, the more conceited and less genial of the Bernoulli brothers (of course being "less genial" than Jakob B does not mean a reproach). Euler was even more genial than both and many others. Nevetheless here he applied the wrong concept.
> > > > >
> > > > > John Gabriel is completely correct when he says:
> > > > >
> > > > > 1. S = Lim S, is wrong

> > > > Of course it is wrong! Euler never wrote that!
> > >
> > > Moron. He did write that.
> > >

> > > > > 2. The series is not the limit.
> > > > Of course not if a series is defined as finite.
> > >
> > > The series in discussion is called an infinite series you idiot. That statement is hand waving bullshit.
> > >

> > > > But if the series is defined as the infinite sum, then it is the limit.
> > >
> > > Wrong. That would be like saying 0.333... the UNIQUE representation of the series is the limit, but point 3 below shows this is not possible.
> > >
> > >

> > > > > 3. 1/3 cannot be expressed in base 10 because 3 is not a prime factor of 10.
> > > > This, and both 1. and 2. is your own invention. Who is the good professor It is WM!
> > >
> > > Oh, so you can express 1/3 in base 10? What a moron!
> > >
> > >

> > > >
> > > > KON

> > > > >
> > > > > Unfortunately the contrary belief has lead to the mess of transfinite set theory.
> > > > >
> > > > > Regards, WM

> >
> > I am completely satisfied with 1/3. I can however do it in base 10.

>
> No you can't. There is a number theorem that states 1/3 is not representable in base 10.


AS a FINITE decimal expansion, yes. You are fucking dense, crank. The theorem does not state there are numbers without an infinite decimal representation. All real numbers have at least one infinite decimal representation.

Den onsdag 27 september 2017 kl. 22:43:10 UTC+2 skrev John Gabriel:
> On Wednesday, 27 September 2017 12:55:42 UTC-4, konyberg wrote:
> > onsdag 27. september 2017 18.05.46 UTC+2 skrev konyberg følgende:
> > > onsdag 27. september 2017 14.20.13 UTC+2 skrev John Gabriel følgende:
> > > > On Monday, 25 September 2017 19:22:51 UTC-4, John Gabriel wrote:
> > > > > https://www.linkedin.com/pulse/part-1-axioms-postulates-mathematics-john-gabriel
> > > > >
> > > > > https://www.linkedin.com/pulse/part-2-axioms-postulates-mathematics-john-gabriel
> > > > >
> > > > > https://www.linkedin.com/pulse/part-3-axioms-postulates-mathematics-john-gabriel
> > > > >
> > > > > https://www.linkedin.com/pulse/part-4-axioms-postulates-mathematics-john-gabriel
> > > > >
> > > > > https://www.linkedin.com/pulse/part-5-axioms-postulates-mathematics-john-gabriel
> > > > >
> > > > > 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

> > > >
> > > > Here is a quote from the only mainstream academic I respect on sci.math:
> > > >
> > > >
> > > > Euler's teacher was Johann Bernoulli, the more conceited and less genial of the Bernoulli brothers (of course being "less genial" than Jakob B does not mean a reproach). Euler was even more genial than both and many others. Nevetheless here he applied the wrong concept.
> > > >
> > > > John Gabriel is completely correct when he says:
> > > >
> > > > 1. S = Lim S, is wrong

> > > Of course it is wrong! Euler never wrote that!
> > > > 2. The series is not the limit.
> > > Of course not if a series is defined as finite. But if the series is defined as the infinite sum, then it is the limit.
> > > > 3. 1/3 cannot be expressed in base 10 because 3 is not a prime factor of 10.
> > > This, and both 1. and 2. is your own invention. Who is the good professor It is WM!
> > >
> > > KON

> > > >
> > > > Unfortunately the contrary belief has lead to the mess of transfinite set theory.
> > > >
> > > > Regards, WM

> >
> > Ok. Some questions for you, JG.
> > S(n) = (i=0 to n)Sum ((-1)^i a^i)), |a| < 1
> > What is this?
> > Can you give me the closed form?
> >
> > And where do this lead to?
> > S = (n=0 to inf)Sum ((-1)^n a^n)), |a| <1
> > Can you give me the closed form of this?
> >
> > Is it the same as the former?
> >
> > Is it: S(n) = S, or lim S(n) = S, or S(n) = lim S(n), or S = lim S. What do you go for, and what would Euler go for?
> >
> > KON

>
> It does not matter how you choose to represent it. The fact is that Euler defined S = Lim S.
>
> S = 1 - a + a^2 - ...
>
> Lim S = Lim_{n \to \infty} (1-(-a)^n) / (1 + a) = 1 / (1 + a)
>
> S = Lim S



No.

S = lim_{n --> infinity} S(n) is not the same as S = lim_{n --> infinity} S.

God, you're fucking dense.


Den onsdag 27 september 2017 kl. 23:37:53 UTC+2 skrev John Gabriel:
> On Wednesday, 27 September 2017 17:21:20 UTC-4, konyberg wrote:
> > onsdag 27. september 2017 23.10.13 UTC+2 skrev John Gabriel følgende:
> > > On Wednesday, 27 September 2017 17:07:02 UTC-4, konyberg wrote:
> > > > onsdag 27. september 2017 22.56.56 UTC+2 skrev John Gabriel følgende:
> > > > > On Wednesday, 27 September 2017 16:47:36 UTC-4, Diogenes Polonium wrote:
> > > > > > On September 27 2017, the genius known as John Gabriel wrote:
> > > > > > > On Wednesday, 27 September 2017 12:41:50 UTC-4, konyberg wrote:
> > > > > > > > onsdag 27. september 2017 18.18.02 UTC+2 skrev John Gabriel følgende:
> > > > > > > > > On Wednesday, 27 September 2017 12:05:46 UTC-4, konyberg wrote:
> > > > > > > > > > onsdag 27. september 2017 14.20.13 UTC+2 skrev John Gabriel følgende:
> > > > > > > > > > > On Monday, 25 September 2017 19:22:51 UTC-4, John Gabriel wrote:
> > > > > > > > > > > > https://www.linkedin.com/pulse/part-1-axioms-postulates-mathematics-john-gabriel
> > > > > > > > > > > >
> > > > > > > > > > > > https://www.linkedin.com/pulse/part-2-axioms-postulates-mathematics-john-gabriel
> > > > > > > > > > > >
> > > > > > > > > > > > https://www.linkedin.com/pulse/part-3-axioms-postulates-mathematics-john-gabriel
> > > > > > > > > > > >
> > > > > > > > > > > > https://www.linkedin.com/pulse/part-4-axioms-postulates-mathematics-john-gabriel
> > > > > > > > > > > >
> > > > > > > > > > > > https://www.linkedin.com/pulse/part-5-axioms-postulates-mathematics-john-gabriel
> > > > > > > > > > > >
> > > > > > > > > > > > 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

> > > > > > > > > > >
> > > > > > > > > > > Here is a quote from the only mainstream academic I respect on sci.math:
> > > > > > > > > > >
> > > > > > > > > > >
> > > > > > > > > > > Euler's teacher was Johann Bernoulli, the more conceited and less genial of the Bernoulli brothers (of course being "less genial" than Jakob B does not mean a reproach). Euler was even more genial than both and many others. Nevetheless here he applied the wrong concept.
> > > > > > > > > > >
> > > > > > > > > > > John Gabriel is completely correct when he says:
> > > > > > > > > > >
> > > > > > > > > > > 1. S = Lim S, is wrong

> > > > > > > > > > Of course it is wrong! Euler never wrote that!
> > > > > > > > >
> > > > > > > > > Moron. He did write that.
> > > > > > > > >

> > > > > > > > > > > 2. The series is not the limit.
> > > > > > > > > > Of course not if a series is defined as finite.
> > > > > > > > >
> > > > > > > > > The series in discussion is called an infinite series you idiot. That statement is hand waving bullshit.
> > > > > > > > >

> > > > > > > > > > But if the series is defined as the infinite sum, then it is the limit.
> > > > > > > > >
> > > > > > > > > Wrong. That would be like saying 0.333... the UNIQUE representation of the series is the limit, but point 3 below shows this is not possible.
> > > > > > > > >
> > > > > > > > >

> > > > > > > > > > > 3. 1/3 cannot be expressed in base 10 because 3 is not a prime factor of 10.
> > > > > > > > > > This, and both 1. and 2. is your own invention. Who is the good professor It is WM!
> > > > > > > > >
> > > > > > > > > Oh, so you can express 1/3 in base 10? What a moron!
> > > > > > > > >
> > > > > > > > >

> > > > > > > > > >
> > > > > > > > > > KON

> > > > > > > > > > >
> > > > > > > > > > > Unfortunately the contrary belief has lead to the mess of transfinite set theory.
> > > > > > > > > > >
> > > > > > > > > > > Regards, WM

> > > > > > > >
> > > > > > > > I am completely satisfied with 1/3. I can however do it in base 10.

> > > > > >
> > > > > >

> > > > > > > No you can't. There is a number theorem that states 1/3 is not representable in base 10.
> > > > > >
> > > > > > State it please...

> > > > >
> > > > > Theorem: Given any rational number p/q with p and q integers, it is not possible to represent p/q in base b unless all the prime factors of q are also prime factors of b.
> > > > >
> > > > > As an exercise, try to prove it! Chuckle.
> > > > >
> > > > > It's very easy. In fact I don't know how anyone can obtain a Math Bsc without knowing this theorem, but aside from Professor WM, even the biggest morons at MIT were not aware of the theorem.

> > > >
> > > > Give a reference to this theorem, except you!

> > >
> > > Any respectable book on number theory has it. Do you know what is number theory? I think not.
> > >

> > > > KON
> >
> > But what you are saying only is true for fractions that are finite in decimal evaluation.

>
> If it's not finite decimal evaluation, then it means you can't express it in that base. There is no such thing as infinite decimal evaluation.
>
> To prove that 1/3 is measurable(representable) in base 10, you need to provide a fraction in the form of k/10^n where k and n are integers. You cannot find such integers.


There is a such thing as an infinite decimal representation. Just because you don't understand limit doesn't mean limits are ill-formed. Your brain is ill-formed.

Den torsdag 28 september 2017 kl. 18:17:11 UTC+2 skrev John Gabriel:
> On Thursday, 28 September 2017 08:07:55 UTC-4, Idiot Jan Burse wrote:
>

> > 1 - a + aa - a^3 + ... taken to infinity will result in 1/(1+a) for |a|<1.
>
> > Of course this is the result when taken to infinity, which
> > means nothing else than getting the limit.

>
> Lesson 1:
>
> In real mathematics, we give objects names. Not just any name like unicorn or crapola, but meaningful names. A name is affected by the concepts involved and the primary attributes. All this is part of well-defining a concept. For example, an orangutan cannot just say: "Define X". Why? Because that tells us nothing about "X". Get it imbecile?
>
> So we give an object of the form a_1 + a_2 + a_3 + ... the name "series".
>
> This means it is a partial sum and the ellipsis means subsequent terms can be determined according so a systematic method (function or rule). It has ZERO to do with "infinity" whatever bullshit that is. So it is called a series. But this is its definition. We instantiate the object by copying the template series and calling it S (similar to what we do in C++ but not exactly the same). We give S the value 1 - a + aa - a^3 + ...
>
> So S is now defined.
>
> Now we notice that S has a special property, that is, it converges as a geometric sequence does to a_1/(1 - r) = Lim {n \to \infty} a_1(1-r^n)/(1-r) |r|<1 where r is the common ratio a_n+1 / a_n. We call a_1 / (1 - r) the LIMIT and Lim {n \to \infty} a_1(1-r^n)/(1-r) we call Lim S. It is some magnitude or distance (neither of which necessarily need to be described by a number, suffice to say that such a magnitude exists. I include this to avoid all the bullshit about rationals, etc).
>
> Notice that Lim S does not depend on knowing ALL the terms of the series, only a general partial sum from which the LIMIT is determined. The LIMIT doesn't even give a fuck if all the terms are there or even there at all!! It only cares about the partial sum to n terms.
>
> So now what Euler does, is he thinks: Jawohl! I vill simply define dis two objects gleich (equal). Because look, the sum cannot exceed or reach the limit. So vy not just make both the same? FAIL!!!!!!!
>
> The one object is a series and the other object is a magnitude/LIMIT.
>
> Orangutan objection 1: The series is a shorthand for the limit. What??!!!! The limit is a number and the series consists of "infinitely many" terms. Chuckle. You fucking hypocritical dumb bastards! Wake up you monkeys!!
>
> Overruled.
>
> Orangutan objection 2: The series is a unique representation for the LIMIT or number (never mind the fact that the limit may not be described by any number). Oopsie. No. When does 3.14159... describe pi? After 10 trillion digits? After 1 billion? No morons, 3.14159... NEVER describes pi. Imagine if you could live forever and all you did was write down the digits of pi. Guess what? You would NEVER complete the task. Do you understand you syphilitic arseholes? 3.14159... is a nonsense concept just like 0.333..., etc. Newsflash: Division is a finite process. It does not carry on forever.
>
> Overruled.
>
> If indeed this were a valid definition, then the sequence 3; 3.1; 3.14; ... would also have a limit that is a rational number.
>
> Orangutan objection 3:
>
> "Of course this is the result when taken to infinity, which means nothing else than getting the limit."
>
> The LIMIT doesn't give a shit about infinity. Infinity does not even feature where the limit is concerned. It is a bogus concept. The LIMIT is deduced from the partial sum to n terms. And of course we do not call the series by the limit because they are NOT the same thing. But that is what naughty Euler did and you baboons have yet to realise.
>
> Overruled.
>
> All your future objections are overruled because you are ALL morons except WM.


We define a new mathematical notation for a limit, and we write a_1 + a_2 + a_3 + ... for lim_{n --> infinity} a_1 + a_2 + a_3 + ... + a_n. A series is not a partial sum, a series *HAS* partial sums. And a series doesn't have to have terms generated by some "rule" or "systematic method". They can be completly arbitrary. And if you are taking the limit, of course you must care about all the terms, because the limit definition requires you to do so. A series is the limit of its partial sums. That's the standrad definition everyone, except fot maybe you, use.


Den torsdag 28 september 2017 kl. 00:53:57 UTC+2 skrev John Gabriel:
> On Wednesday, 27 September 2017 18:47:39 UTC-4, John Gabriel wrote:
> > On Wednesday, 27 September 2017 18:19:35 UTC-4, Me wrote:
> > > On Wednesday, September 27, 2017 at 1:07:35 PM UTC+2, John Gabriel wrote:
> > >

> > > > This is what you wrote moron!
> > >
> > > You really should visit a doctor, John.
> > >
> > > Hint: No, I didn't write the following, phycho:
> > >

> > > > Lim S = 1/(1 + a) = S = 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... is
> > > > defined to be Lim S = lim_{n to infinity} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n.

> >
> > You did write that Psycho. All I did was label the parts and show you that your God Euler wrote it. My article is the end of your bullshit for good. You will have to live out the rest of your life in shame for not noticing the ill-formed definition.
> >
> > https://www.linkedin.com/pulse/eulers-worst-definition-lim-john-gabriel
> >
> >
> > S 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ...
> >
> > Lim S is just short for
> >
> > lim_{n->oo} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n.
> >
> > The latter is 1/(1 + a) (for |a| < 1 that is).
> >
> > Hence we may write (as Euler did):
> >
> > Lim S = S or S = Lim S

>
>
> For any imbecile to write
>
> that 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ...
>
> is just short for
>
> lim_{n->oo} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n.
>
> is to deny that 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... exists because
> lim_{n->oo} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n is NOT the same and denotes only the limit, not the partial sums, which are presumed to be infinite in the mythical infinite series.
>
> What morons are mainstream orangutans.



But they are the same, because we define the series to BE the limit of its partial sums.

Den torsdag 28 september 2017 kl. 01:09:11 UTC+2 skrev John Gabriel:
> On Wednesday, 27 September 2017 19:04:14 UTC-4, John Gabriel wrote:
> > On Wednesday, 27 September 2017 18:57:53 UTC-4, Me wrote:
> >

> > > Hint: "idée fixe has also a pathological dimension, denoting serious psychological issues"
> >
> > Of course. It is primary a result of brainwashing. I have often claimed that teaching imbeciles like you is much harder than teaching those who have not been infected with the same rot as you.
> >
> > It is true that false knowledge is harder to unlearn than it is to learn true knowledge.
> >
> > Euler indeed wrote S = Lim S which is of course a very bad idea that you cling to.
> >
> > I would recommend a psychologist, but it would do your syphilitic brain no good as you are past the point of no return.

>
> In order to get a quick introduction, I suggest you watch my one minute video.
>
> Euler made a mistake in defining S = Lim S. No matter how hard modern academics have tried to defend this ill-formed definition and to deny it most recently, the truth is hard to refute and stands out.
>
> Page 91 and 92 of Euler's Elements of Algebra is clear and irrefutable evidence that Euler defined S = Lim S.
>
> S = 1 + 1/2 + 1/4 + 1/8 + ...
>
> Lim S = 2
>
> If we continue the series (S) to infinity, there will be no difference at all between its sum (infinite sum), and the value of the fraction 1/(1-a) (* Lim S), or 2.
>
> It is just incredible how academics of the BIG STUPID (mainstream academia) have denied this consistently. The most recent denier was David Ullrich from OK state university.
>
> It is a fact that Euler defined S = Lim S.
>
> (*) The limit.
>
> =====================================================
>
> Without much ado, here is the original German text with my translations and commentary following:
>
> 295.
>
> 1+1/2+1/4+... ohne ende. Denn nimmt man nur zwei elieber, so hat man 1+1/2, und so fehlet noch 1/2. Nimmt man drei elieber, so hat man 7/4, fehlet noch 1/4: nimmt man vier elieber, so hat man 15/8, fehlet noch 1/8: woraus man sieht, das immer weniger fehlet, folglich, wenn man unendlich weit fortgeht, so mus gar nichts fehlen.
>
> 1 + 1/2 + 1/4 + ... without end. If one takes two terms, then one has 3/2, and so 1/2 still remains. If one takes three terms, then one has 7/4 with 1/4 remaining. If one takes four terms, then one has 15/8 with 1/8 remaining, from which one observes that which remains is less each time. Consequently, if one proceeds infinitely, nothing remains.
>
> COMMENTARY: Notice that Euler believed that one could proceed infinitely, that is, *add up all the terms* into a 'infinite' sum. He truly believed that eventually nothing would remain. This was Euler's first reference to an infinite sum.
>
> 296.
>
> Man sehe a = 1/3, so wird unser Bruch 1/(1-a) = 1/(1-1/3)=3/2, welchem daher folgende Reihe gleich ist 1+1/3+1/9+... bis ins unendliche. Nimmt man zwei elieber, so hat man 4/3, fehlet noch 1/6. Nimmt man vier elieber, so hat man 40/37, fehlet noch 1/54. Da nun der Fehler immer dreimal fleiner wird, so mus derselbe endlich versdwinden.
>
> If a = 1/3, then we have 1 / (1-a) = 1 / (1-1 / 3) = 3/2, So the series for this fraction is equal to 1 + 1/3 + 1/9 + ... when taken to infinity. If one takes two terms, one has 4/3 with 1/6 remaining. If one takes four terms, one has 40/37 with 1/54 remaining. Since the error always decreases, it must eventually vanish.
>
> COMMENTARY: In 296, Euler elaborates further on the process taken to infinity with the same conclusion, that is, all the terms are summed and nothing remains - it all vanishes.
>
> It is very clear that Euler believed in an infinite sum. In today's mathematics, we know that 1 + 1/3 + 1/9 + ... has a limit of 3/2. But Euler wasn't happy just to think of it as an upper bound, he dogmatically stated that the sum is indeed equal to the limit. And he again states this candidly in 296.
>
> Euler calls his series (Reihe) 1 + 2/3 + 4/9 + ... (S) and the sum (or limit as we know it today Lim S) 3. Euler equates these two objects in 298, that is, S and Lim S.
>
> 298.
>
> Daher ist unser Bruch 1/(1+a) gleich dieser unendlichen Reihe:
>
> 1 - a + aa - aaa + aaaa - ...
>
> Hence the fraction 1 / (1 + a) {Lim S} is equal to this infinite series:
>
> 1 - a + aa - aaa + aaaa - ... {S}
>
> COMMENTARY: In 298, Euler leaves no doubt that S equals Lim S as he states this clearly, and not just with examples as one sees in the previous cases.
>
> And this is the evidence which cannot be refuted. Only a ignorant, dishonest and incompetent academic will still scream and shout.
>
> Euler's original text can be found here:
>
> http://reader.digitale-sammlungen.de/...
>
> Conclusion:
>
> Some may even try to argue that the definition is well formed. This is quickly dismissed by the fact that infinity is a junk concept which can't be reified in any way whatsoever.
>
> Till this day we have fallacies such as 0.333... = 1/3.
>
> This fallacy is quickly dismissed by knowledge of the following number theorem:
>
> Given any fraction p/q, then an equivalent fraction can be found in base b, if and only if, all the prime factors of q, are also factors of b.
>
> On that theorem alone, it is mind-boggling how anyone can claim 0.333... = 1/3.
>
> Let's move along and think about the limit of the series 0.3+0.03+0.003+...
>
> It can be written as follows:
>
> S = 3/10 + 3/100 + 3/1000 + ... + 3/10^n + 1/3 x (1/10^n)
>
> But the tail part, that is, 1/3 x (1/10^n) is chopped off and we are left with a supposedly "infinite" series:
>
> S = 3/10 + 3/100 + 3/1000 + ... + 3/10^n + ...
>
> S = 1/3 [1 - 10^(-n) ] + ...
>
> Lim (n -> oo) S = Lim (n -> oo) 1/3 [1 - 10^(-n) ] + ... = 3/10 / (1 - 1/10) = 1/3
>
> As n -> oo, it is clear that the right hand side above will have 1/3 as its limit.
>
> So according to Euler, S which is equal to 3/10 + 3/100 + 3/1000 + ... + 3/10^n + ... is also equal to Lim (n -> oo) S which is equal to 1/3. That is, S = Lim S.
>
> It is very easy to see that the limit is 1/3. But to equate the series to its limit?! That's absolutely senseless. Academics might claim that at infinity, the sum will be 1/3, but as we've seen, the number theorem rejects that. Hence, it's not possible to represent 1/3 in base 10, as anything else besides a rational number approximation.
>
> The next retort is that 0.333... is only a symbol. Well, this is ridiculous because 1/3 is very well defined and needs no other representation. Besides, the question arises of what does 0.333... mean. If it means the limit, then that is quite absurd, because the limit is well defined, that is, 1/3. If the representation is of chief consideration, then the only way to get 0.333... is to imagine a bogus infinite sum. The ellipsis does not mean an infinite sum, nor does it mean all the 3s are there. In fact, the limit 1/3 does not care if the 3s are all there or even there at all!
>
> In the same light, one can't write 3.14159... and call it pi. This too is meaningless nonsense. Consider that no rational number approximation of pi is ever equal to the measure of that incommensurable magnitude pi, which is represented by measuring a circle's circumference using the diameter as a unit. Pi is not a number.
>
> These bad ideas and definitions have snowballed, so that we have nonsense such as 0.999... = 1 and a bogus mainstream calculus. Rather than correct and revise mathematics, the orangutans who sit atop the academic trash heap simply continue on the same path that has resulted in zero progress the last 150 years.
>
> For the first and only rigorous formulation of calculus in human history, you will have to discard infinity, infinitesimal, limit theory, real numbers and any other ill-formed definitions.
>
> To learn much more than you have learned in all your school years, visit my YT Channel. Also learn about the 13 fallacies in mainstream mathematics.
>
> The following dishonest academics in the BIG STUPID, have constantly libeled and belittled my claims:
>
> Gilbert Strang, Professor of mathematics - gilstrang@gmail.com (MIT)
>
> Jack Huizenga, Professor of mathematics - huizenga@psu.edu (HARVARD)
>
> Anders Kaesorg, PhD student - andersk@mit.edu (MIT)
>
> David Ullrich, Professor of mathematics - david.ullrich@math.okstate.edu (David Ullrich)
>
> I am certain they would love to hear from you with any questions you might have and that is why I have included their email addresses. Also, I am certain you will want to know why they are still holding onto these fallacies.
>
> Some possible handwaving responses you might get from these academics:
>
> Mythmatician: But Euler meant the limit.
>
> No, Euler did not mean the limit. Euler did not say that the fraction 1/(1+a) is the limit and it's very clear he did not think of it as the limit, by the statements:
>
> 1. The remainder vanishes.
>
> 2. The infinite sum (S) is the fraction (Lim S).
>
> To be certain, there is no S without adding up the terms, and to those who are silly enough to argue 0.333... is just the unique representation, well, consider that you can't have 0.333... without adding the terms. When is it unique? After one trillion terms? One light year of terms? Infinity?! Afraid not, infinity is a junk concept.
>
> Mythmatician: There was no formal word for limit in German.
>
> Nonsense. The German language had enough words back then (*) to describe the idea, even if there was no official or formal word for limit. Besides, the word Grenz (border) was known in Euler's time. Euler was smart enough to use it, if he really meant to say limit.
>
> (*) The German nation had just descended from living in the trees only a couple of hundred years before Euler wrote his Elements of Algebra, which are rather primitive when compared with the Works of Archimedes written thousands of years before Euler.
>
> Mythmatician: But every possible marker/point is covered in the interval (0,1).
>
> Stupid academics imagine that every point is covered in a given interval and so an infinite sum to them is possible. But this is disproved by Archimedes in this eye-opening video less than 2 minutes. Archimedes used the Archimedan property of rational numbers (not real numbers because there aren't any!) and proof by contradiction to show that the area of the parobolic segment is 4/3 the area of the triangle on the same diameter/chord. Archimedes never recognised any other numbers besides the rational numbers.
>
> Professor W. Mueckenheim made this comment on sci.math:
>
> Euler's teacher was Johann Bernoulli, the more conceited and less genial of the Bernoulli brothers (of course being "less genial" than Jakob B does not mean a reproach). Euler was even more genial than both and many others. Nevertheless here he applied the wrong concept.
>
> John Gabriel is completely correct when he says:
>
> 1. S = Lim S, is wrong
>
> 2. The series is not the limit.
>
> 3. 1/3 cannot be expressed in base 10 because 3 is not a prime factor of 10.
>
> Unfortunately the contrary belief has lead to the mess of transfinite set theory.
>
> And this comment on the irrationality of pi:
>
> It is strange that this clear and concise argument is always circumvented and only irrelevant details are discussed.
>
> (1) The union of the sequence of intervals [0, (n-1)/n) is [0, 1).
>
> The limit of the sequence of intervals [0, (n-1)/n] is [0, 1].
>
> (2) The union of the sequence of finite initial segments of the sequence of natural
>
> numbers {1}, {1, 2}, {1, 2, 3}, ... has less then aleph_0 elements.
>
> The limit of this sequence (if existing) has aleph_0 elements.
>
> (3) The sequence of partial sums 3.1, 3.14, 3.141, ... = 3.141... is not irrational.
>
> The limit of this sequence is irrational.
>
> It is strange that is always circumvented. Really? Is it?
>
> Let's summarise a few reasons why it is a bad idea to define S = Lim S:
>
> [i] It leads to non-equations such as 1/3 = 0.333...
>
> In algebra, we use the equality sign between numbers. One might say that 0.333... is the number 1/3, because it represents the limit of the series 0.3+0.03+0.003+..., but the problem with this approach, is that academics misguidedly try to perform "infinite" arithmetic using 0.333... and arrive at further absurd results such as 1 = 0.999... To say 0.333... is the limit, is like decreeing that 1/3 = 0.333... In mathematics there is no place for rules or decrees, only logic and common sense.
>
> [ii] Many academics get the wrong idea that 0.333... is actually an infinite sum, which is obviously impossible. Then their colleagues will deride them by claiming that it's not an infinite sum, only a representation of 1/3. Well, one cannot arrive at this representation without the fallacy of infinite sum. The representation is not a result of long division because long division is a finite process. There is also confusion among academics about the Euclidean algorithm and the long division algorithm. These are not the same!
>
> [iii] A number theorem in mathematics states that given any p/q and base b, it is not possible to represent p/q in that base b, unless b contains all the prime factors of q. In order to claim that 1/3 can be represented in base 10, you need to find an m and n, such that 1/3 = m / (10^n), where m and n are both integers. Good luck! One would think that on this theorem alone, academics would have been smart enough to realise that S = Lim S is a very bad idea indeed.
>
> [iv] This leads to numerous other wrong ideas with respect to set theory and is a major time waster with no practical application in science, technology or engineering. The only numbers ever used by humans are the rational numbers. There is no such thing as a "real" number. It is an illusion and a myth.
>
> Confusion in the mainstream
>
> But why has the mainstream never realised this blunder you might ask. Well, before you ask this, you may be surprised to know that mainstream academics do not even understand the advocates of this wrong theory, to wit, Rudin on page 59 of his third edition repeats exactly what Euler stated, that is, S = Lim S.
>
> Rudin's analysis textbook has been the De Facto real analysis textbook used in most courses on real analysis. There is also an irony in the name "real analysis" because there is no valid construction of real number.



"At infinity" is usually understood as a LIMIT. Namely, a series is the limit of its partial sums. Not the limit of itself. It's the limit OF ITS PARTIAL SUMS.


Den torsdag 28 september 2017 kl. 01:14:25 UTC+2 skrev John Gabriel:
> On Wednesday, 27 September 2017 19:08:49 UTC-4, Me wrote:
> > On Thursday, September 28, 2017 at 12:53:57 AM UTC+2, John Gabriel wrote:
> >

> > > to write
> > >
> > > that
> > >
> > > 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ...
> > >
> > > is just short for
> > >
> > > lim_{n->oo} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n.
> > >
> > > is to deny that 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... exists

> >
> > Huh?! John, it's getting worse with you.
> >

> > > because lim_{n->oo} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n is
> > > NOT the same

> >
> > Huh?!
> >
> > Look, idiot, if I use the expression "1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ..." to DENOTE
> >
> > lim_{n->oo} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n

>
> You can't do this you infinitely stupid ape! They are not the same thing. The former is what you misguidedly call an "infinite series" and the latter is the "limit" of that infinite series.
>
> Baboon!!!!! You have an obsession with S = Lim S because you were brainwashed with it.
>

> > then
> >
> > 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... = lim_{n->oo} 1 - a + aa - a^3 + a^4 - a^5 + a^6 - a^7 + ... + a^n
> >
> > BY DEFINITION.

>
> Huh? What are you saying moron?!!!!! Do you realise what you have just written? Obviously not. You first say that you are denoting it by Lim S and then defining it to be itself. Get a grip you idiot!
>
> You are desperately trying to save the indefensible. The most honourable thing for you to do is admit you are wrong and stop pretending to be an authority because it is very unbecoming of retards.
>
> <Just too much wrong with the rest that follows.>


We define an infinite series to BE the limit of its partial sums. They are the same. PER DEFINITION.

Den torsdag 28 september 2017 kl. 17:03:09 UTC+2 skrev John Gabriel:
> On Thursday, 28 September 2017 08:07:55 UTC-4, burs...@gmail.com wrote:
> > Am Donnerstag, 28. September 2017 13:55:40 UTC+2 schrieb John Gabriel:
> > > False. Newton thought that 1 - a + aa - a^3 + ... taken to
> > infinity would result in 1/(1+a) for |a|<1.
> >
> > Of course this is the result when taken to infinity, which
> > means nothing else than getting the limit.

>
> What?!! Nothing can be "taken to infinity" you monkey!!!
>
> There is no such thing as infinity. I will keep reminding and warning students of mathematics against your syphilitic brains.
>

> >
> > Why am I so stubborn, and not able to understand that phrases
> > like ad infinitum, taken to infinity, &c., etc... are nonsense?

>
> Because you are a moron!


Taking something to infinity usually means to take a limit approaching infinity.

Den torsdag 28 september 2017 kl. 19:43:46 UTC+2 skrev John Gabriel:
> On Thursday, 28 September 2017 13:20:24 UTC-4, burs...@gmail.com wrote:
> > Nope, the equal sign is between values.
>
> Wha?!! You imbecile! That's what an equal sign means! S is a series and its value according to Euler is Lim S, so naturally S = Lim S. YOU MORRROOOOON!!!
>
>

> > Both sides of the equation sign have the meaning of a value.
> > An equal sign between values and sequence wouldn't
> > make any sense, since in set theory, usually this

>
> Oh! So you mean S = S ?? Chuckle. You fucking idiot. If that were the case, there would be no need to have an equal sign in the first place.
>
> It stops RIGHT HERE!
>
> <too much shit!!!!>



S, which is the value of the series, is defined as the limit of S(n), the nth partial sum, as n approaches infinity. The series IS NOT defined to be its own limit, because S does not depend on n.