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Topic:
Euler's Blunder S = Lim S : 12/7/2017 7:06
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1
Last Post:
Dec 7, 2017 7:11 PM




Re: Euler's Blunder S = Lim S : 12/7/2017 7:06
Posted:
Dec 7, 2017 7:11 PM


You forgot:
If you made a mistake, dont be affraid to correct it.  Konfuzius
Am Freitag, 8. Dezember 2017 01:06:42 UTC+1 schrieb John Gabriel: > https://www.linkedin.com/post/edit/6209800972248715264 > > > In order to get a quick introduction, I suggest you watch my one minute video. My most recent video and probably the last on S = Lim S, is around 5 minutes. > > Euler made a mistake in defining S = Lim S. No matter how hard modern academics have tried to defend this illformed definition and to deny it most recently, the truth is hard to refute and stands out. > > Given S_series and S_sequence, it makes no difference how you interpret the Eulerian Blunder. All of the following are illformed definitions: > > S_series = Lim S_series > > S_series = Lim S_sequence > > S_sequence = Lim S_series > > S_sequence = Lim S_sequence > > Example: > > S_series = 0.3+0.03+0.003+... > > S_sequence = { 0.3; 0.33; 0.333; ... } > > 1/3 = Lim S_series = Lim S_sequence > > As you can see, the limit is the same and every one of those interpretations leads to the same conclusion: S = Lim S is an illformed definition. > > 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/(1a) (* 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/(1a) = 1/(11/3)=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 / (1a) = 1 / (11 / 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.digitalesammlungen.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 mindboggling 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 illformed 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 eyeopening 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, (n1)/n) is [0, 1). > > The limit of the sequence of intervals [0, (n1)/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 nonequations 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.



