If it requires getting to infinity, it cannot be reached.
No. the >limit< does not “go to” anything. The summation goes toward the limit, as indicated by “as x → infinity”, not “when x = infinity”.
A “limit” for an infinite series is a mathematical expression referring to a value that is progressively approached yet never reached.
Consider the following two infinite series:
Σ[9/(10^n)] = .9 + .09 + .009 + …
Σ[1/(2^n)] = .5 + .25 + .125 + …
lim Σ[9/(10^n)] = .9 + .09 + .009 + … = 1
lim Σ[1/(2^n)] = .5 + .25 + .125 + … = 1
Both of those have the same limit = 1. Yet the second is provably less than the first.
That is like saying that 90% of the way to the top of the mountain is as close as is possible to get, thus the top of the mountain is only 90% of itself.
It is a “proper decimal expansion”. And it says, “you can’t completely represent that fraction in decimal because there can never be a final end”.
On the other hand, infinity is only a mathematical concept which may or may not have corresponding physical reality.
As such, conceptually it may be equally maintained whether infinity is an open or closed concept. Reality can not furnish an example where a string can be infinitely divided so as to accomodate the conceptual limit of 1 , between 0 and 1
Yes, but isn’t that just a practical restriction? We can’t write infinite decimal places, granted. But we know that if .33333… continued on infinitely, it would equal 1/3, right? There’s no contradiction saying we can’t completely write it out and saying that an infinite string would equal 1/3. We don’t need to know the final end of pi to solve equations in terms of pi. We don’t need to know the final end of sqrt(2) to know that squrt(2)^2=2. What’s the problem?
The problem is , that at present accuracy of utilization, there is no problem. But at the rate of acceleration of technological progress, and future need for exponential need for more and more accuracy, the requirements of carrying the decimal much farther may carry the decimal point accordingly.
Therefore, the conceptual application may for a very long time trump the empirical approximations.
The applications may trail the mathematical requirements for
reasons of having to do with expected or , projected requirememts.
That is false.
n/infinity = undefined infinitesimal.
The fact that they converge at different rates tells you that one is a lesser cardinality than the other (albeit fractional).
Each and every 1/(2^n) that is summed into the accumulation is less than each 9/(10^n), thus there is no possibility for the second sum to ever accumulate as much as the first - infinitely.
Just after the first 10 summations, we have:
0.9999999999
0.9990234375
And it can never catch up.
Plus we have the fact that more than half of the trailing digits for every partial sum throughout the entire infinite set are less than 9, with no opportunity for any digit to ever be more than 9 (leading to an infinite list of digits that are less than 9 as a part of the infinite set of digits). If merely one out of the entire infinite set of digits is less than 9, the entire sum must be less than the sum of all digits being a 9. The fact that the series of digits never ends means that the trailing half of less-then-9 digits are ALWAYS there. Thus the entire infinite series represents a number than is necessarily less than the first series.
No. It is a restriction in concept too. It is saying that there will always and forever be a remaining amount that has not been included. The ellipsis symbol means that an end result CANNOT be reached and thus IS NOT available in this digital form.
"Infinitely"does NOT mean that you get to an end “at infinity”, because there is no “at infinity”.
It means there is no end to finally reach the conclusion, thus there is always an infinitesimal remaining amount.
That is true for WHY it shouldn’t be taught incorrectly (as it currently is). The same is true for a great deal of physics that is good enough in a practical sense (just as thinking that the Earth is flat), but isn’t exactly right and thus will lead to conflict, confusion, and perhaps danger in the future. Why teach a lie? What is being gained?
That’s why the sum of each infinite series is 1: for any (standard) real number x, there is an m such that the sum s from 1 to m is between 1 and x. The only way that that can be true is if the sum of the infinite series is equal to 1.
Again, it sounds like you’re talking about a limit. The ellipsis doesn’t mean we’re waiting for it, hoping that eventually enough decimal places will build up underneath it that we can wave our hands and move on. The value in every decimal place is specifically and statically defined by the ellipsis (or r or bar or several other conventional shorthands for ‘like this infinitely’).
Just like we cross an infinite number of real numbers when we count 1, 2, 3…, we imply an infinite number of decimal places with .333… They’re all there, they aren’t reaching for anything. They’re all already full of 3s.
It seems as though you want to imply that at least one of them is not a 3, but that’s not the case. They are all 3s, so there is no infinitesimal amount remaining.
I can’t tell if that even makes sense.
For each sum from 1 to n for series (1), S1n, there is an m such that: 1 > S2m > S1n
If I’m reading that right, it is false. S1n > S2m, for n=m, always. And for n<>m, it is irrelevant.
Of course I am talking about the limit. And the ellipsis is telling you that the string of 9s cannot ever reach that limit. It really isn’t merely a shorthand issue. The shorthand is short for expressing an infinite series, but it is NOT shorthand for “When it finally gets to infinity” - and that is how you are reading it.
There is no “getting to infinity” even in concept.
No. I am saying that there is ALWAYS more 3s required to get to the limit. There CANNOT ever be enough 3s, even in concept because in concept there is no end to infinite (there is no “infinity” to be reached). It is impossible to have enough 3s.
If 256/infinite is the whole equation then the result is exactly 0.
To not be 0 it would have to be multiplied by infinite.
Then the result could still be 0, it could be infinite or it could be any Real value like 5.
Such a case requires a different look at the equation at hand and then it may be possible to determine the value.
Some basic rules which are always true -
infinite/infinite = not defined (could be anything)
(infinite^2)/infinite = infinite
infinite/(infinite^2) = 0
infinite*0 = not defined (could be anything)
Or let’s take integral calculus - Here infinitesimally small elements are summed up infinitely often and the result can be calculated. It will be 0, or infinite or a defined value, depending on the equation (function).
If the equation is simply 17.7898/infinite then the result is exactly 0.
What I just wrote down is basic bitch math. If you have an issue with it then you have an issue with the consensus in mathematics and they are pretty conservative when it comes to their definitions and how all the various operations and theories are interlinked.
Unfortunately for the bluster of this paragraph, I think James is correct on this point. As I understand it, 1/infinity is not a proper mathematical formula, because infinity is not a number. You can take the limit of 1/x as x goes to infinity, and that’s zero, but it’s improper to treat infinity as a number and plug it in directly for x.
Moreover, I don’t think it’s obvious that the limit x^2/x as x goes to infinity is infinity. My understanding is that, for a countable infinity, a plane that countably infinite in two dimensions has the same size as a countably infinite 1 dimensional line. Each point in the coordinate plane can be uniquely assigned to exactly one number on the infinite line. That means that there are an equal number of points on the line and the plane.
I’m saying that for any finite sum of the one series, there is a finite sum of the other series that is closer to 1. Even if one series will always lag behind, it will also always have a finite sum that is closer to 1 than any finite sum of the other series. For that reason, it can’t be said that one infinite sum is strictly larger than the other. The speed at which they converge does not seem relevant to the value of the infinite sum.
To put it another way, if we plotted the values of each, even though the curves would be different, the area under each curve would be exactly 1.
I don’t see the distinction you’re trying to make here.
But there are infinite 3s, there can’t be more 3s (again, dealing with a countably infinite string of decimal places). There doesn’t need to be an end, and in fact an end would make the string finite and so less than 1/3. But an infinite string has all the decimal places it needs: all of them.
The shorthand is for infinite series but not for getting to 1. The shorthand is particularly a mathematical notation, and not based on assumptions of notating an infinite function limited by 1 , but on assumption of there being such a limit. (Since the limit by
definition is qualified by an infinite function).
There is no “in infinity” just as there is no “at infinity”.
Infinite is a quality, not a quantity. The quality is that of being “endless”.
What you are saying is:
1/endless = 0
256/endless = 0
“endless” isn’t a number that can be divided into anything, nor is “infinity” because it is not a quantity.
We aren’t “basic bitches” here. We don’t give credit for being right to the world merely because was there before we were. And this isn’t high school. We are examining analytical perfection, not practical applications taught to the world in general.
Ack! Your right. I was thinking of the partial sums, not the individual terms … My bad
It still doesn’t catch up. It just isn’t so obvious.
That’s irrelevant. “Each runner eventually gains more distance than half as much as the other.” Well, yeah, so what.
The “area under the curve” IS the “summation”. They are the same thing. So you are just repeating your false assertion with different words.
.
That … that … that is the issue and problem.
It is YOU, YOU, YOU who are presuming a destination of “enough 3s” to finally equal 1/3. That point does not exist. There is no “enough 3s” to satisfy the stipulation. “Endless” or “infinitely” MEANS that there is no satisfied point to get to. There is no “infinity” or end to “endless”.
The convergence must either be satisfied or not, right?
If it could be satisfied, it would equal 1/3 (in this case).
But the ellipsis is conveying that the series cannot be satisfied - ever.
You are right about that: infinite divided by 11.599999999…. = still infinite
and because it’s not a number, 1/infinite = any number divided by infinite = zero.
Right again, more like advanced bitching.
You seem to use the term world in the sense of society or people at large.
So, what have you learned from examining analytical perfection, found any chinks? Or poles? The 1?
An infinitesimally small element has no size, it’s not defined.
In a math-world built around abstractions, that’s what nothing is - it’s not defined, a cypher.
Let’s say two different series are moving not towards a Real number but towards infinity, at different rates though -
Would you say they eventually end up at different infinities?
Because that would be the inversion of such a difference between two infinitesimally small nothings.
0.99999…. + 0 = 1
I find that your thinking here is very common though. Not gonna say it’s ‘basic bitch’ because that sounds negative to your ears.
You seem to think in a way like - “Well, there is some number and that number gets smaller and smaller but it doesn’t get reduced to zero, in no step, thus, at the end, there must still be some-thing.” - True, except for infinite.
Cardinality of different number spaces?
I’m talking about something like your proposed two series, both in the Real number space.
Like, do you think
lim x->infinite f(x)= 9*x
a different infinite (the function value) than
lim x->infinite f(x)= x
?
I don’t think so but then again maybe a google search can find me another theory for some other dimensional constructed number space.
As for absolutes, they can be found in the thinking of man.
Since when is not being able to measure something an issue for mathematics?
Anyway, I’ve looked it up on wikipedia where the first introductory sentence talks about how it’s too small to measure. Historically that’s based on physicists or an analogy to a physical problem.
I guess, the nice thing about things which are not defined, it leaves room for your own creations.
And something doesn’t have to get to infinity to identify what it is converging towards. Math is not reality, it’s an idealised abstraction. Within it absolutes can be and are defined. Zero.
