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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2
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1answer
37 views

A Puzzle on Infinity: How to guess the color of hats? [duplicate]

Infinitely many (i.e. $\omega$ - many) people each have either a white hat or black hat on their heads. Each person can see everyone's hats except their own. Each person simultaneously announces a ...
1
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1answer
36 views

Inverse of a matrix with main diagonal elements approaching infinity

Let $A$ be a invertible, symmetric, positive definite $p \times p$ covariance matrix with main diagonal elements $a_{ii},~i = 1,~\ldots,~p$. If all main diagonal elements would approach $\infty$, ...
43
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16answers
3k views

Different sizes of infinity

Correct me if I'm wrong, but this is what they taught us in precal: $$\lim_{x\rightarrow\infty}x=\infty$$ $$\lim_{x\rightarrow\infty}x^{2}=\infty$$ But, we also know that $n^{2}>n$ if $n\notin [0,1]...
2
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1answer
94 views

What's so different about limits compared to infinitesimals?

If you find the limit is 2 for a given function, wouldn't this be the same as $2 + \epsilon$ with $\epsilon$ being a negligible value? This different way of defining limit-like behavior seems rigorous ...
-1
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0answers
30 views

Can the sum of a series be used unconditionally as that value? [closed]

I am in a debate with someone. We are working a convergent series and in order to answer a related question the series has to be exactly equal to that sum (there is no margin for error, has to be that ...
0
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3answers
105 views

How $\infty=\infty$.

If we contruct two strainght lines as shown: Then join them such that to complete a triangle. It is taught that we can find infinity points on straight line. So there are infinity points on $DE$ ...
2
votes
1answer
66 views

Topology and Borel sets of extended real line

Let $\mathcal{B}_{X}$ denote the Borel $\sigma$-algebra on $X$. I'm reading a book on real analysis by Folland and he defines $$\mathcal{B}_{\overline{\mathbb{R}}} = \{ E \mid E \cap \mathbb{R} \in \...
0
votes
1answer
51 views

What are the limit points of $A_n=[n,\infty)$ in a metric space? Is $A_n$ closed?

$A_n=[n,\infty)$ in $\mathbb{R}$ with a Euclidean metric. A set is closed if it contains all its limit points. A limit point is a point whose neighborhood contains a point in the set. I'm not sure ...
0
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2answers
32 views

Infinite differentiability with a removable discontinuity?

I'm still a beginner with calculus. But this puzzled me. Let's say you had $f(x) = \frac{x^2-1}{x+1}$. It's discontinuous at one point. If you took the derivative infinitely many times, would the ...
2
votes
1answer
53 views

Is this sequence going to infinity, and how do we know that?

$a+\dfrac {a+\dfrac {a+\dfrac {a+\dfrac {:} {b}} {b}} {b}} {b}=?$ I've tried letting $\quad a+\dfrac {a+\dfrac {a+\dfrac {:} {b}} {b}} {b}=K$ Which makes the equation: $a+\dfrac {K} {b}=K$ $\quad$ ...
2
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1answer
87 views

Proof of Lemma 8.2.3 in Terence Tao Analysis 1 book

$\textbf{Lemma 8.2.3 }$ Let X be a countable set, and let $f:X \rightarrow R$ be a function. Then the series $ \sum_{x \in X} f(x)$ is absolutely convergent if and only if $$ sup \left\{ \sum_{x \...
4
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5answers
150 views

Why can't we just say that $\infty-\infty$ equals zero?

Let be $\lim\limits_{x\to \infty}x=A$ and $\lim\limits_{y\to \infty}y=B$. Can be $A-B=0$? If the answer is "no" , why? And my other example: $\displaystyle\int_{-\infty}^{\infty} \dfrac{x}{...
0
votes
3answers
62 views

Where did $\sqrt{x^2/x^2}$ come from in $\lim_{x \to -\infty}\frac{x+1}{\sqrt{x^2}} = \lim_{x \to -\infty}\frac{-1-1/x}{\sqrt{x^2/x^2}} = -1$?

I'm reading a calculus book and I saw the following limit solution. $$ \lim_{x \to -\infty}\frac{x+1}{\sqrt{x^2}} = \lim_{x \to -\infty} \left(\frac{x+1}{\sqrt{x^2}} \cdot \frac{-1/x}{-1/x}\right) = ...
2
votes
1answer
53 views

What is meant by $\lim_{x\to \infty^+}$

I am familiar on how limits work and such. For example, look at the following limit: $$\lim_{x\to 5^+} \frac{-x^2+5x}{5-x} = 5$$ It is saying that, as $x$ approaches $5$ from the right, the equation ...
0
votes
1answer
32 views

Compute H-infinity norm in Matlab

Please can someone write a command in Matlab for calculating $H_{\infty}$ norm for the following system: $$\frac{d}{dt}z(t)=Az(t)+Bu(t)+Fw(t)$$ $$y(t)=Cz(t)+Du(t)$$ where $A$, $B$, $C$, $D$, and $F$ ...
2
votes
2answers
76 views

Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
0
votes
2answers
41 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
82 views

Set theory with multiple countable infinities [closed]

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 ...
-3
votes
2answers
63 views

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

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....
0
votes
1answer
32 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
869 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
75 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
47 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
votes
0answers
29 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 $y\to\infty$...
1
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1answer
46 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 \infty}...
1
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3answers
50 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 ...
5
votes
4answers
186 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
41 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
15 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
votes
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
30 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 $b_1$), $\lim_{(...
3
votes
1answer
32 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
54 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
71 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
42 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 ...
-1
votes
1answer
57 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
38 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
140 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
votes
1answer
42 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
vote
3answers
33 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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votes
1answer
63 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 non-...
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
26 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
vote
2answers
73 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 ...
119
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
vote
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, $B$ ...
1
vote
1answer
30 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!