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Re: An infinite sum is NOT a limit and 0.333... is not well defined as 1/3.
Posted:
Oct 5, 2017 10:11 AM


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. 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.
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. This is the definition we use, and you must show the definition does not work. You cannot just ignore reality and the fact that we actually define infinite sums in this way. 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.
All limits are not defined as limits of finite sums. Limits are defined for example with an (?, ?)definition and does not depend on the notion of sums at all. An infinite sum is a particular kind of limit.
So we can't literary add an infinite amount of terms. That is correct, and no one ever said we could. However, we can define a such "sum" in terms of a limit. Since the partial sums of a decimal expansion of a real number will form a real Cauchy sequence, it will converge to an unique element in ? (since limits are unique). So hence all decimal expansions will be a real number. The converse is also true, all real numbers will have a decimal expansion. This expansion will not be unique though. 1.000... and 0.999... is the same real number.
And a sequence is a function. It isn't "derived" from a series, whatever that means. We use the notion of sequences in order to define series. A sequence can be constant, and can be a divergent Cauchy sequence. This is easy to prove.
A real number is not defined as a limit of some rational Cauchy sequence, nor it is a rational Cauchy sequence itself. We define a real number as an equivalence class of rational Cauchy sequences given a particular equivalence relation on the set of all rational Cauchy sequences.
We can define a concept without knowing how to calculate it. If you claim that the limit definition is circular, you must show it. But your argument is not an argument, because nothing in the definition requires us to explicitly know the limit.
I could define p to be the 568458678677547866785676746665443343437886878467:th prime. Is this definition flawed because I do not know what p is? I can show that p will be welldefined and exist, so why isn't this a flawed definition?



