You can’t accept both statements (“Anything times zero is zero” and “Anything times infinity is infinity”) since they are in opposition to each other (accepting them both would be a logical contradiction) so you have to pick one.
If you accept that “Anything times zero is zero” then zero times infinity is zero.
If you accept that “Anything times infinity is infinity” then zero times infinity is infinity.
Either way, (lim_{n\to\infty} \frac{1}{n} + lim_{n\to\infty} \frac{1}{n} + lim_{n\to\infty} \frac{1}{n} + \cdots) is not equal to (1). It’s either (0) or it’s infinity.
And the correct answer is (0) (since infinity times zero, properly speaking, without deviating from the standard meaning of the term zero and/or the standard meaning of the term multiplication, is zero, not infinity.)
But that’s no proof that infinite series that converge do not exist (Ecmandu’s point.)
For an undefined, there isn’t really a positive or negative argument, maybe a neutral one.
But for the thread topic as a whole, there’s a singular defined answer, so you can start conversation about it. Somebody did and math ended it several times over throughout this thread.
You accept one preliminary statement: it’s undefined.
You accept this because yes - you can’t pick more than one defined conclusion when they’re mutually incompatible, even when they’re seemingly just as valid as each other.
“Picking one” singular defined conclusion → invalid (no more or less valid than any other, for now at least).
“Picking one” undefined conclusion is the only option left at this point.
Wrong.
You only mention two of the possible “general rules” that I mentioned. Another one was that any number multiplied by its reciprocal is 1. So yes, there is a perfectly valid rule that can also be seen as applying to this situation, which equals 1 like the tendency did all the way through all possible finite numbers. This is all the extra work you need to do in this case to actually resolve what would otherwise be “undefined” as the answer, and it’s just the same as you would do in calculus (as I explained in a previous post to iambiguous). Just like in calculus where limits of 0 and infinity seem to pose conceptual issues along the way to an answer, the answers you end up with are still perfectly valid, exact and unique, you can get past the same conceptual issues in the same way for Ecmandu’s argument as well.
So you begin rejecting 1 singular defined answer over any other that’s seemingly just as valid, investigate this undefined answer to see if more work can yield a singular defined answer, it can - so that’s the correct answer: 1. That’s the correct, full process. You don’t stop short and conclude there’s a problem with math as soon as you come across what might seem like a problem with math.
The step of converting the problem to the form “(\lim_{n→\infty}\frac1n+\lim_{n→\infty}\frac1n+\lim_{n→\infty}\frac1n+⋯)”, or “(\lim_{n\to\infty}(n)\times\lim_{n\to\infty}(\frac1n))” just turns out to be a red herring, when you could have just simplified the problem to (\lim_{n\to\infty}(\frac{n}n)) or just (\lim_{n\to\infty}(1)), which is quite obviously 1 like it was for all finite numbers for n. Trying to turn it into something that looked like it would suddenly switch to equalling 0 at its limit just created more work, which at best was only useful for showing how easy it is to manipulate infinity to giving arbitrary undefined answers. If you’re careful with them, you can often get past undefineds in a legitimate way.
Don’t simply claim this is wrong. Calculus has been doing this and getting correct answers for 350 years.
So in Ecmandu’s example, his infinite series does converge - to 1.
In the thread topic, there’s also a convergence of what can be represented as an infinite series ((0.\dot9)) also to 1.
It’s a logical contradiction to say that both statements are true. They can’t be both true. So which one is true? You didn’t answer this question. Is it “Anything times zero is zero” or is it “Anything times infinity is infinity”? It’s either one or the other, you can’t have it both ways.
Yes. The third rule says that (n \times \frac{1}{n} = 1) for any number (n). I ignored it because it didn’t look particularly relevant.
The argument put forward by Ecmandu is, at least as I see it, based on the premise that (1 = \lim_{n\to\infty} \frac{1}{n} + \lim_{n\to\infty} \frac{1}{n} + \lim_{n\to\infty} \frac{1}{n} + \cdots). This premise can be restated as (1 = \infty \times \lim_{n\to\infty} \frac{1}{n}).
The three general rules you mention are relevant because they decide how we’re going to evaluate the right side of the equation.
Rule #1: Anything times zero is zero.
If we accept it then the right side of the equation is (0) which is how both me and Ecmandu evaluate it. This leads to (1 = 0) which is a contradiction.
Rule #2: Anything times infinity is infinity.
If we accept it then the right side of the equation is (\infty). This is not how we evaluate it, but even if we did evaluate it this way, we’d end up with (1 = \infty) which is still a logical contradiction.
Rule #3: Anything times its reciprocal is one.
This rule doesn’t seem relevant. This is because there is an underlying, unspoken, assumption that zero is the reciprocal of infinity. So perhaps, it is better stated as “The reciprocal of infinity is zero”.
Thus, (1 = \infty \times \lim_{n\to\infty} \frac{1}{n}) is equal to (1 = \infty \times 0) and since (0) is the reciprocal of infinity this leads to (1 = \infty \times \frac{1}{\infty}) which is (1 = 1). Thus, no logical contradiction.
The question is:
Do we accept the premise? I personally don’t. Zero is not the reciprocal of infinity and this is evident in the fact that zero times infinity isn’t (1).
(The third rule can also be stated as “Zero times infinity is one” which makes it more similar to the other rules.)
It can be examined as a convection within differing sets. The difference between the nominal minima appears as irreducible within a matrix of minima: the smallest supposed " god" particle to the cosmological infinite phenomenal set.
Mathematical analysis trends toward the primary logical approach.
But both minima and maxima is generalized within the approach to integrate toward infinity:
I got your point from the start, don’t worry about me.
What I’m trying to explain is that when it comes to infinity, you can make anything mean anything if you play around enough. This doesn’t mean everything does actually mean everything else, and anything equals nothing etc., it just means that infinity is a slippery concept to deal with.
With your example, you use infinite infinitessimals, which you decided to present as meaning zero only, when it can mean anything - that’s what would make it undefined - but that’s why I said that this means you need to do more work. It doesn’t mean “oh just stop prematurely at the first controversial defined answer”. And it only means stop at “undefined” if more work can’t be resolved to a specific “defined” answer. And we can resolve this case, meaning infinite series aren’t all undefined. Only the divergent ones are. The convergent ones are defined.
I explained that we know through the pattern of finites that tend towards the infinite limit, and the fact that you can simplify earlier on before introducing limits at all, to where it’s obvious that this pattern for finites does actually hold for the infinite limit situation as well (and doesn’t suddenly change out of nowhere at the limit). Just that extra little bit of extra work shows the answer is actually 1=1 throughout in the exact same way that calculus has always worked.
What wasn’t clear about my direct addressing of this very part?
“Anything times zero is zero” and “Anything times infinity is infinity” lead to defined solutions. I explained that because you can’t pick a single defined solution when the others are just as valid in this specific case, and you can’t pick multiple mutually incompatible defined solutions like you said, the only valid option remaining would be to pick the singular “undefined” solution. But this is only preliminary, as I also said.
I then explained that when you get to an undefined, you can do more work to find out if there is actually a singular defined solution afterall. We note that you can simplify the equations in the way I pointed out, which matches the finite tendency, and just like we do for calculus, we can safely conclude that 1 does actually equal 1 afterall, even after the attempt to try and make it look like it equalled 0.
There is no “having it both ways here”. It’s just what you’re supposed to do for a sufficient investigation that doesn’t stop short for dramatic effect and false glory.
And no surprises, when forced to look at the most relevant piece of the puzzle, you dismiss it as irrelevant.
Quite obviously (\frac{n}n=1) for all finite values as n tends towards infinity. The numerator tends to infinity, the denominator tends to zero - I don’t care what you think about the reciprocal of infinity - the tendencies are clear, and they don’t even need to come into play if you just simplify (\frac{n}n) to (1) to begin with. The whole manipulation to make it look like 1 is anything other than 1 by taking advantage of the complications of infinity is just a red herring.
This is the answer by the way, whether you understand/accept it or not. Math has exact unique answers, unlike most disciplines.
I wouldn’t pretend any of those other disciplines are as open/shut as this, but this is math and it is not woolly and open for debate. It’s the opposite, and I wouldn’t be like this if it were any other way.
There is no debate here, there is only learning the correct solution or continuing to impotently deny/dismiss it as irrelevant. If you want debate, try another topic that isn’t math. If you want to learn, learn.
A direct response to that part of my post would be one (and only one) of the following two statements:
“Anything times zero is zero” is a true statement (the other is false)
“Anything times infinity is infinity” is a true statement (the other is false)
You didn’t pick your choice.
Which one of the two statements is true? They can’t both be true.
I don’t care what these general statements lead to. I am not talking about “solutions” whether they are “defined” or “undefined”. I am talking about the two general statements themselves. They can’t both be true. Only one of them can be true. That’s basic logic. So which one is true? You didn’t answer this question.
It isn’t. You have to understand the question before you can determine whether what you provided is truly an answer. It doesn’t look like you understood the question. Nothing wrong with that, of course, but it has to be pointed out in order to direct the conversion in the right direction.
Here’s my understanding of Ecmandu’s argument. Of course, this is merely a guess, so it might be quite wrong. This will be so until Ecmandu learns how to communicate (most likely never to happen, so it’s probably for the best to simply ignore him and let him whine.)
The limit of (\frac{1}{n}) as (n) goes to infinity is (0).
(1 = 0 + 0 + 0 + \cdots)
1 = infinity times zero
(1 = 0)
Since we already know that (1 = 0) is false, and since we know that the above argument is logically valid, it follows that some of our premises are false.
Which one(s)?
Ecmandu thinks it’s premise #4.
I think it’s premise #3.
It would be great if Ecmandu could verify this to be an accurate representation of his argument.
I did feel as if I addressed this point when you raised it earlier. The real number 1 and the integer 1 are used for very different purposes. That horse is just a bunch of quantum fields, it’s not really a horse. Physicists tell us that.
Any specialized field of knowledge has conceptual frameworks that seem like nonsense at the every day level.
It’s an algorithm. It’s also a sequence but note that it’s a very specific kind of sequence. Namely, it is a sequence of instructions. On the other hand, (\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dotso) is a sequence of numbers. So it’s not an algorithm. The distinction is subtle but it’s real.
The fact that the output of your algorithm is a sequence of numbers does not mean that that sequence of numbers is an algorithm.
Man I’m late to this party… but it seems to me this has more to do with language and difinitions than any mathematical “proofs”.
You have to conclude given the definition of infinity is literally “unquantifiable” then the difference between 1 and 0.999… is “unquantifiable” which is another way of saying you cannot describe the difference… and if you can’t describe the difference… well… they are then going to have to share the same description.