Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.

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3answers
32 views

Can one non-cardinal infinity be greater than other non-cardinal infinity?

As far as I know, there are two different notions to the word "infinity" in Mathematics. First notion of infinity has to do with the cardinality of a set: if a set contains infinite number of ...
1
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1answer
67 views

Set theory with multiple countable infinities [on hold]

In set theory, all sets that are countably infinite are generally considered to have the same size since there is a bijection between them. Has anyone tried formalising set theory in a way which ...
-2
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2answers
61 views

If $\dfrac{1}{\infty}=0$ then I can prove that $0 = 1$ [on hold]

Given, $\dfrac{1}{\infty}=0$, then $1=0 \cdot \infty = 0$ (because $0$ times any number or values is $0$ and here that number is infinity). Which gives us $1=0$ i.e, $0=1$. Hence proved....
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1answer
31 views

Voronoi edges example

I have 4 line segments: 0 0 2 0 // 1st line segment 2 0 2 1 // 2nd line segment 2 1 0 1 0 1 0 0 and I wrote some CGAL code to print the Voronoi edges. However, ...
3
votes
5answers
847 views

How many points in a line segment?

My teacher said that in the circumference of circle there are infinite points. When I was learning more about circle, I came to this picture: My question is: When we unroll the circle, then the ...
0
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0answers
37 views

Is there an infinity smaller than countable? [duplicate]

In other words: is $\aleph_0$ the smallest infinity? Is it easy to prove?
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2answers
71 views

Random Room changing in the Hilbert hotel. [closed]

Let's say you have a Hilbert's grand hotel full occupancy. Assign each occupant a new room select randomly without regard to whether the room is assigned to someone. i.e. empty rooms, multiple ...
4
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0answers
43 views

Flea on the coordinate system

We drop a flea on a point of the coordinate system(with integer coordinates). Due to the dimensions of the flea we can not see it. The flea jumps away every second by one unit (always in the same ...
0
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0answers
27 views

Complex variable limit at infinity

Is $\lim\limits_{z\to\infty} \frac{4z^2}{(z-1)^2}$, $z\in\mathbb{C}$, evaluated the same way as a real variable function limit? Or does one need to show separate cases for $x\to\infty$ and ...
1
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1answer
44 views

Limit of $(-1/2)^n$ as $n$ approaches infinty

I tried plugging bigger and bigger $n$'s into my calculator and the result obviously approaches $0$ (albeit oscillating between positive and negative). So how do you prove that: $$\lim_{n \to ...
1
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3answers
48 views

Arithmetic Operations with Infinities in Real Analysis

Infinity is not a number , thus we cannot perform the usual arithmetic operations that we do with real numbers This is the usual reason given when asked why we can't perform the usual arithmetic ...
6
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4answers
165 views

First year calculus student: why isn't the derivative the slope of a secant line with an infinitesimally small distance separating the points?

I'm having trouble with the limit approach to calculus ever since I heard about the infinitesimal definition. Maybe you can help me settle what's been bothering me this year. Looking at the limit ...
0
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0answers
40 views

Difficulty in understanding Cantor's diagonal argument

I recently found Cantor's diagonal argument in Wikipedia, which is a really neat proof that some infinities are bigger than others (mind blown!). But then I realized this leads to an apparent paradox ...
0
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0answers
14 views

Average of left and right limits | Signum function, Heaviside step function, and Grandi's series

This question probably already has an answer but usually involves stuff that's way over the top of my head so I'm hoping for a simple explanation. In Adams, R. A., & Essex, C. (7th edition) ...
0
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1answer
45 views

What is the formal definition of a limit at infinity?

I keep coming across two different kinds of answers to this question. The first definition: We say that $$\lim_{x\to \infty} f(x) = L$$ if the following condition is satisfied: for every number ...
3
votes
1answer
29 views

How do I calculate this limit when two terms tend to infinity at similar rates

In a particular problem that I am currently trying to solve, I have the following expression (this is not the entire expression, I have included only the terms involving $a_1$ and $b_1$), ...
3
votes
1answer
31 views

Interval notation: infinity, -infinity in closed interval

I was watching a video stream a little bit ago and noticed on an equation without context that had the interval $\left[{-\infty, \infty}\right]$. This was preculiar to me as I've never seen the ...
1
vote
1answer
50 views

precise definition of a limit at infinity, application for limit at sin(x)

(a) Write down the first principles definition of the statement $\lim\limits_{x→∞} f(x) = L$. For this I have that for every $ε >0$, there is a corresponding number $N$, such that if $N>0$, ...
1
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2answers
70 views

did i use infinite wrong?

This algebra question is in Dutch and the original file van be found here: Question 19 Ill try to translate the important info needed to answer this question. $$s= \frac{(a+b)} { (ab)}$$ S= dpt ...
1
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2answers
41 views

Will a decreasing probability ever resolve favorably? [duplicate]

Let's say I start off with a 50/50 chance at winning the lottery. But I lose. Now my chance is only half as good, or 25%. I lose again. Now the chance is 12.5%. Same result. If this continues all ...
0
votes
1answer
52 views

Find a **bijection** between two intervals

I am struggling with this question and was hoping somebody could help me, Thanks Find a bijection between the intervals $(-1,1)$ and $(0,4)$ where $X \in R$
2
votes
3answers
36 views

Division of segments into infinitely many parts.

Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2. If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested ...
0
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0answers
91 views

When adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?

I was watching Ramanujan: Making sense of 1+2+3+... = -1/12, where the presenter writes: (I tried to write this out in $\LaTeX$ but couldn't figure out how to do multi-column alignment without ...
2
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1answer
40 views

Summing Over Uncountable Index Sets

In answering the question Why do we classify infinities in so many symbols and ideas?, William's answer asserted that summing over an uncountable index set necessarily results in an infinite sum. I am ...
1
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3answers
28 views

Does the graph $y=\sin(x)\times\sin(x^{-2 })$ cross the $x$ axis an infinite amount of times in a finite interval?

Vsauce made a video recently on counting past infinity, and he represented the set of natural numbers to infinity with a set of lines, where each successive line is a smaller distance away from the ...
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1answer
58 views

What is infinity to the zeroth power? [closed]

I am not happy with the answers posted to similar questions. For example, in: What is infinity to the power zero the accepted answer is 1, which is definitely wrong. I think the answer is any ...
1
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2answers
43 views

Can I subtract infinity from infinity?

I was stuck when solving a problem on limits. It was like----> $\lim_{x\to\infty} (x-x)$. What should I do now?
28
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7answers
3k views

Why do we classify infinities in so many symbols and ideas?

I recently watched a video about different infinities. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, ...
0
votes
2answers
25 views

Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
1
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2answers
71 views

How to define $[-\infty, \infty]$ or $[0, \infty]$?

I am familiar with basic undergraduate topology. For example, I know the process of one point compactification of a non-compact topological space, and how it applies to, say, $\mathbb R^2$. My ...
118
votes
12answers
7k views

Are we allowed to compare infinities?

I'm in middle school and had a question (my dad is helping me with formatting). We're learning about infinity in math class and there are a lot of problems like how it's not a number and how if you ...
1
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1answer
10 views

Let $A$ be an infinite set and let $B$ be a set such that $A$ is equinumerous to a subset of $B$. Then, $B$ is infinite.

To me, the proof is as simple as this: Let $C\subset B$ such that $A\sim C$. Then, as $A$ is infinite, we have that $C$ is infinite. Thus, as $C\subset B$, it must be that $B$ is infinite. Thus, $B$ ...
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1answer
29 views

Find the value of the Infinite product in terms of k which is a positive integer

$$\prod_{n=k+1}^{+\infty}\left(1-\frac{k^2}{n^2}\right)$$ The only help we have been able to find is that of Euler, anything would be amazing!
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2answers
92 views

Infinite product of negative numbers? $-1\times -1\times-1\times -1\dots=$ [closed]

Edited: Making the question as brief as possible to avoid future confusion and misunderstanding. Note This was moved as a separate question from: Product of all real numbers in a given interval ...
3
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1answer
89 views

Why can't we keep adding axioms forever?

Let F be a formal system falling prey to Gödel's incompleteness theorems, implyng there is a true but unprovable statement, call it $G_1$. Of course, adding $G_1$ to the axioms of F doesn't solve the ...
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3answers
90 views

Definition of limit as $x\rightarrow \infty$

Every time i get confused with the definition of $\lim_{x\rightarrow \infty}f(x)=L$. I could not find a reference that will give the definition. I am trying to write what i understood. See if this is ...
0
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2answers
49 views

Show that if |f(x)| converges in infinity, so is f(x).

I think that in a I should compare the function |f(x) - f(x) and 2|f(x)| but I am not sure how i would do that. Also, I am not sure how i should duduce what i want to deduce in b after i find a.
10
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3answers
331 views

Product of all real numbers in a given interval $[n,m]$

READ-ME I have now what I can call for myself answers to all my problems and subquestions proposed in this post, thus I accepted Strings answer as the answer to this question since it was of most ...
0
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1answer
15 views

How to compute log likelihood for impossible events?

I am defining a set $\mathbf{Z} = [p,q,r,s]$ such that $Pr(p)+Pr(q)+Pr(r)+Pr(s)=1$. Likelihoods are defined as follows \begin{align} \lambda_p&=\log \frac{Pr(x=p)}{Pr(x=s)}, \hspace{2mm} ...
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0answers
3 views

How do we accomplish the subtraction of two infinities in a PWL Approximation?

I am trying to implement a piecewise-linear function of an M/M/1 Queueing system in an ILP to approximate the delay values. I have expressed my PWL constraint as follows: $\alpha_{i}+ \beta_{i}u_{n_s} ...
6
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3answers
820 views

Would an infinite random sequence of real numbers contain repetitions?

If random real numbers are selected from the set of all real numbers, for an infinite number of iterations, what is the likelihood of repetitions occurring?
1
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1answer
27 views

Finding percentage when infinity is involved

Is it possible to convert a function of the form $f(x)=ax/(a-x)$ to a form where you can find $f(x)/f(a)$? I'd like to find the percentage of $f(a)$ for $f(x)$ but this seems impossible while $f(a)$ ...
1
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2answers
24 views

Dichotomy in the number of regions on a plane formed by an infinite number of lines

I'm reading Knuth's Concrete Mathematics and we are dealing with recurrence relations. He proves that the number of regions $L_n$ formed by $n$ lines on a plane is $L_n=\frac{n(n+1)}{2}$. I don't ...
0
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2answers
67 views

Concept of infinity: Infinity - Infinity

What solution does $\int_0^\infty 1 dx - \int_a^\infty 1 dx $? yield if (i) $a\in (0, \infty)$ and (ii) $a=0$? From the Continuum hypothesis I concluded that each integral is uncountable infinite ...
0
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2answers
116 views

How does it equal -1/12? [duplicate]

So all my friends keep telling me that if you add up all the numbers from 1 to infinity, (1+2+3+4...) then the answer is -1/12. They showed me this proof with infinite sums, but I didn't understand ...
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3answers
1k views

Why is cardinality of set of even numbers = set of whole numbers?

I recently watched a YouTube video on Banach-Tarski theorem (or, paradox). In it, the presenter builds the proof of the theorem on the basis of a non-intuitve assertion that there as as many even ...
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0answers
20 views

Let $f, g$ be defined on $(a,\infty)$ and $\lim_{x \to \infty}f(x)=L$ and $\lim_{x \to \infty}g(x)=\infty$, then $\lim_{x \to \infty}f(g(x)) = L$

If $\lim_{x \to \infty} g = \infty$, then for $M>0$, there exists $d_{1} >0$ such that if $x>d_{1}$, then $g(x) > M$. If $\lim_{x \to \infty} f = L$, then given $\epsilon>0$, there ...
1
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3answers
91 views

Finding $\sqrt {6+\sqrt{6+\sqrt{6+…+\sqrt 6}}}$

For positive integer $n$, $$x_n=\sqrt {6+\sqrt{6+\sqrt{6+...+\sqrt 6}}}$$ where $6$ is written $n$ times. How can we find the $x _\infty$ ? I coded a program any found that $x _\infty$ would be ...
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3answers
110 views

Is $\lim_{x \to -∞} (2+3x)^{2/3}$ positive or negative? [closed]

$\lim_{x \to -∞} (2+3x)^{2/3}$ Is this $(-∞)^{2/3} = (-∞^2)^{1/3} = +∞$ ?
0
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4answers
56 views

Finding limit without using $(a-b)(a+b)$ method

I'm working on this problem: $$\lim_{x\to \infty} (\sqrt {x^2 + 2x} - \sqrt {x^2 - 4x})$$ I tried the following approach and currently it's wrong: $$\sqrt {x^2 + 2x} - \sqrt {x^2 - 4x}$$ Taking out ...