Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: An infinite sum is NOT a limit and 0.333... is not well defined as 1/3.
Posted:
Oct 5, 2017 8:15 PM


Den torsdag 5 oktober 2017 kl. 19:17:21 UTC+2 skrev John Gabriel: > On Thursday, 5 October 2017 10:11:19 UTC4, Markus Klyver wrote: > > Den onsdag 4 oktober 2017 kl. 21:08:51 UTC+2 skrev John Gabriel: > > > On Wednesday, 4 October 2017 14:41:08 UTC4, Markus Klyver wrote: > > > > We define the sum ? (a_i) from i=0 to ? > > > > > > as the SERIES. See, if I populate the subscript i, I get: > > > > > > a_1 + a_2 + a_3 + ... > > > > > > which is what you have below. > > > > > > > Euler defines the SERIES above as its limit of ? (a_i) from i=0 to n as n approaches ?. > > > > > > a_1 + a_2 + a_3 + ... = Lim ? (a_i) from i=0 to n as n approaches ?. > > > > > > > Under this definition 0.333... is indeed 1/3. > > > > > > So is 0.abcd... indeed 1/3 if anyone is stupid enough to define it that way. > > > > > > > This is a DEFINITION of a very useful shorthand notation. > > > > > > > > a_1 + a_2 + a_3 + ... is an infinite sum, and defined AS A LIMIT. > > > > > > It is either an infinite sum or a limit but NOT both. Of course it is strictly speaking a FINITE SERIES because there is no such thing as an INFINITE SERIES. > > > > > > > > > > > a_1 + a_2 + a_3 + ... is not the same as a_1 + a_2 + a_3 + ... + a_n and a_1 + a_2 + a_3 + ... is not "a very long but finite sum". > > > > > > Agreed. > > > > > > a_1 + a_2 + a_3 + ... is a FINITE SERIES that consists of a partial sum with an ellipsis following it. The ellipsis indicates that if an index is given, the corresponding term can be determined. > > > > > > > > > > > a_1 + a_2 + a_3 + ... is an infinite sum, which is DEFINED to be lim_{n > ?} (a_1 + a_2 + a_3 + ... + a_n). > > > > > > Repeating a lie won't make it seem to be true. Chuckle. > > > > > > > All infinite sums are defined as limits of finite sums, > > > > > > But if infinite sums are limits, then you are saying: > > > > > > "All limits are defined as limits of finite sums" > > > > > > Which is nonsense. > > > > > > > because you can't really add an "infinite amount of terms". > > > > > > But if you can't add an infinite amount of terms, then how do you get to represent all mythical "real" numbers as infinite strings of decimals? Chuckle. > > > > > > How do you get 0.333... or 0.999... or 3.14159... or any one of the other infinite strings? See what a fucking moron you are? > > > > > > > Which is why we DEFINE INFINITE SUMS AS LIMITS. > > > > > > Which is why Euler defined S = Lim S > > > > > > > > > > > Also, sequences can be constant and a sequence is usually defined as a function from the natural numbers into the set we're interested in. > > > > > > All sequences are derived from series. Yes. > > > > > > > And no, not all Cauchy sequences are convergent. > > > > > > Wrong. ALL Cauchy sequences without any exception converge to some limit. > > > > > > > It depends on the structure you are considering. > > > > > > Nonsense. In the case of "real numbers", you don't even have a structure to converge to because there are no real numbers and you are using a Cauchy sequence of rationals which you then flippantly call part of an equivalence class. > > > > > > > And no, nothing in the limit definition requires us to know what the limit is. > > > > > > Wrong. All limit definitions require prior knowledge of the limit. This is especially true in the case of the derivative. > > > > > > > We can perfectly define limits without knowing how to prove limits or evaluating limits. > > > > > > Wrong. > > > > > > > > > > > And no, we don't define real numbers as limits of Cauchy sequences. > > > > > > Yes, you do! All the sequences of an equivalent Cauchy sequence have one thing in common  the limit. > > > > An infinite series *is* a limit. Per definition. > > Of course per definition  an illformed definition S = Lim S as I have explained to you over and over again. > > But in fact, a SERIES is not the same as a LIMIT. You defining an apple to be an orange, does not make the apple an orange. Do you understand this? It seems to me that you are blinding yourself because you know that once you admit this error, all your mythmatics falls apart  calculus, set theory, real analysis. And of course this is my goal  to destroy the rot you and other fools have built. It needs to go because I say so and I am infinitely more intelligent than ALL of you combined. I AM the greatest mathematician alive today. That is not even questionable any longer. Three words are the clue: The New Calculus. > > Not a single dick came close to realising it before me. Not even a variation or anything similar. It is a feat of amazing human genius. And here you are, an absolute moron arguing with me about something I have disproved conclusively. How do you think that makes you look? > > > You fail to understand this. We DEFINE the sum ? (a_i) from i=0 to ? as the limit of ? (a_i) from i=0 to n as n approaches ?. And under this definition, we DO have 0.333... = 1/3. > > S = Lim S. > > I define Markus Klyver = Genius. > > Does that mean Markus is really a genius? I think not. Chuckle. > It is a very bad idea to define a series to be equal to its limit. Euler would be ashamed if he were alive today. > > > > > By stating "it is either an infinite sum or a limit but NOT both", you show you don't know what you are talking about. An infinite sum is not an actual infinite sum of infinite many terms. An infinite sum is interpreted and defined as the limit of the partial sums. > > > > a_1 + a_2 + a_3 + ... is not a finite sum. a_1 + a_2 + a_3 + ... + a_n is a finite sum. There's a difference. a_1 + a_2 + a_3 + ... is an infinite sum, which is defined as the limit of the finite sum a_1 + a_2 + a_3 + ... + a_n as n approaches ?. > > > > You can't just deny a definition. > > I have never denied the definition you idiot! I have stated it clearly: S = Lim S. I reject it as ill formed. Yes, I can reject it as nonsense. I am an authority. You are nothing. > > > This is the definition we use, and you must show the definition does not work. > > I have done so in so many ways you retard. Over and over again. Prof. WM agrees with me and you are not anywhere near his stature. He is a mainstream academic like you! > > > You cannot just ignore reality and the fact that we actually define infinite sums in this way. > > I have never ignored it. I have pointed out your ignorance and stupidity. I piss and shit on your definitions. You are inferior to me. I will decide what is a wellformed definition, not you and the collective! Get it? Do you think that because the orangutan council thinks it is well formed, that I must agree? > > The argument is not whether it has been defined that way or not. It's clear that the definition is S = Lim S. Always has been. The argument is that it is ********ILL FORMED*********. Try to understand this you baboon!!!!! > > > Any textbook on analysis will have this definition, rational analysis as well as real analysis. And in any context where infinite sums is dealt with. > > Of course, even Rudin agrees it is S = Lim S. > > Does that make it well formed? Of course not. > > > > We can define a concept without knowing how to calculate it. If you claim that the limit definition is circular, you must show it. > > Has been shown. > > > But your argument is not an argument, because nothing in the definition requires us to explicitly know the limit. > > S = Lim S requires that you know the moron you imbecile! LIM S is the LIMIT. > > > > > I could define p to be the 568458678677547866785676746665443343437886878467:th prime. Is this definition flawed because I do not know what p is? > > That right there just proves you do not understand what is a definition. You don't get to define numbers as nth primes  they are either demonstrably primes or not. No definitions are involved except the definition of prime number in general. > > > I can show that p will be welldefined and exist, so why isn't this a flawed definition? > > Not even wrong.... Sigh..... You have a long, long way to go.
And you keep stating "S = Lim S". I honestly don't know what you mean by it, because it's so abusive and ambiguous. Is S a space? A sequence? A number? What does Lim mean? Is it a functional? Some other undefined thing? You haven't explained any of this.
A series IS a limit, because we define a series as a limit. You haven't explained why we can't make a such definition. We define the value of the series as the limit of its partial sums. Note that the series IS NOT its partial sums. It's the limit of its partial sums.
So S = \lim_{n \to \infty} S(n).
where S is the value of the series and S(n) the nth partial sum.
This is NOT the same as So S = \lim_{n \to \infty} S, which you seem to believe.
And no, we don't need to know how to calculate limits in order to define them. And we don't need to know what the 568458678677547866785676746665443343437886878467th prime is in order to define p as the 568458678677547866785676746665443343437886878467th prime. This is ridiculous. We are not defining some natural number p to be the 568458678677547866785676746665443343437886878467th prime. We are defining p TO BE the 568458678677547866785676746665443343437886878467th prime.
If I let z = x + y, do I need to know how to calculate x and y in order to have z welldefined? No, I only need to know x and y both exist in a sense addition is welldefined between them. I can define p to be the 568458678677547866785676746665443343437886878467th prime, and I can define a limit without knowing how to calculate either of them. I can define L to BE a limit. I don't say, "take this real number L and now it's a limit". I say: "if there is a real number ? which we call call L ? that satisfies the limit definition, I am going to call it the limit".
How is any of this so hard for you to understand?



