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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3answers
42 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
votes
0answers
170 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
46 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 ...
-4
votes
1answer
66 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
vote
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
votes
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
30 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
75 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 ...
120
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
11 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
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1answer
31 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!
-1
votes
2answers
101 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
votes
1answer
98 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 ...
5
votes
3answers
91 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
votes
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
votes
3answers
368 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
votes
1answer
16 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} \...
0
votes
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
votes
3answers
839 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
vote
1answer
30 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
vote
2answers
25 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
votes
2answers
78 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
votes
2answers
124 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 it,...
11
votes
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 ...
1
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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
vote
3answers
92 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 ...
-2
votes
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
votes
4answers
58 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 ...
0
votes
0answers
41 views

Does log(aleph-null) have any meaning?

I'm familiar w/ the meanings and derivations of $\aleph_0$ and the general consequences of the continuum hypothesis (and the discussions at this question. ) So, if it turns out that $2^{\aleph_0} = \...
1
vote
1answer
36 views

Is $\sum_{n=0}^\infty (a \cdot r^n)$ equivalent to $\lim_{n \to \infty}\sum_{k=0}^n (a \cdot r^k)$?

In other words, when writing down an infinite sum, are we always implying that it's actually the limit of that series as the number of terms approaches infinity, or is there some subtle difference?
1
vote
1answer
154 views

Is this line of reasoning correct/valid?

I'm only in the second month of my first calculus course, so I'm not sure how much sense this question will make. I'll give it a try anyways though. Let's say you have the sum of an infinite series ...
0
votes
1answer
51 views

How can one prove this generalization?

In two dimensional space, the length of a vector is $$\sqrt{x^2+y^2}$$ In three dimensional space, the length of a vector is $$\sqrt{x^2+y^2+z^2}$$ How can one prove that in n th dimensional space ...
0
votes
1answer
34 views

A simple question about limits.

This may seem like a simple question, but I feel as if it is wrong but I am unsure why. Is it possible to evaluate a limit in two stages for example: say you know that $x(1- a)\rightarrow b$ as $x\...
1
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0answers
31 views

Applying a general function an infinite number of times

I am trying to learn more about infinite application of functions and functionals. My background is in quantum chemistry, so please forgive some of my notation and terminology. Motivation In ...
2
votes
4answers
89 views

Help finding $\lim_{x \to \infty} {(1 + e^x)}^{e^{-x}}$

$\lim\limits_{x \to \infty} {(1 + e^x)}^{e^{-x}}$ Here are the steps I have taken so far : $\ln{L} = \lim\limits_{x \to \infty} \ln{({1 + e^x})^{e^{-x}}}\\ \ln{L} = \lim\limits_{x \to \infty} {e^{-x}}...
0
votes
2answers
69 views

Positive continuous function with non-zero limits in $\pm\infty$ whose integral over $\mathbb{R}$ is $1$?

Is it possible to create a positive continuous function with non-zero limits in $+\infty$ and $-\infty$ whose integral over $\mathbb{R}$ is $1$? I am studying the probability density functions and ...
1
vote
2answers
67 views

$\infty$ and $-\infty$ are to $\aleph_0$ / $\beth_0$ as “what” is to $\beth_1$?

So, I recently asked a question about whether $\beth_1$ had a negative, and I was promptly reprimanded because I confused $\aleph_0$ with $\infty$. Therefore, to help me understand the concepts ...
1
vote
3answers
36 views

Simplifying a $\lim_{x\to\infty}$ problem.

So I have a problem regarding limits in my calculus class: $$ \lim x\rightarrow\infty \frac {(1+2x^{1/6})^{2016}}{1+(2+(3+4x^6)^7)^8}$$ Basically what I've identified is that it's an $\frac{\infty}{\...
1
vote
2answers
43 views

limit of fraction with factorials

I am trying to take the limit of the following fraction : $$ \lim_{N \to\infty} \frac { N !}{(N-r)!} $$ Attempts : I tried using the Stirling approximation $\ln(n!) =n \ln n - n $ but I figured it ...
1
vote
1answer
64 views

Is the Cartesian product of two countably infinite sets also countably infinite?

I am trying to determine and prove whether the set of convergent sequences of prime numbers is countably or uncountably infinite. It is clear that such a sequence must 'terminate' with an infinite ...
0
votes
2answers
49 views

Infinite limit of trigonometric function

I'm trying to find the limit of a trigonometric function as x approaches $\infty$ so I can't use the fact that : $$\lim_{x\to \infty} \frac{1}{x} = 0$$ For example this limit : $$\lim_{x\to \infty} \...
2
votes
2answers
94 views

Proof of $+\infty=-\infty$ (Maybe)

I guess we can agree that $+0 = -0$. Now, after that, I was simply looking at some graphs. The graph of $\tan x$ shows asymptotes at x = $n\pi + \pi/2$. I got to thinking, what if they weren't ...
-2
votes
3answers
73 views

What is $\lim\limits_{x \to \infty} \dfrac{x}{x-1}$? [closed]

I want to calculate the limit $$ \lim\limits_{x \to \infty} \dfrac{x}{x-1}. $$ I know that this can be achieved using l'Hospital but cannot figure out how to do this.
1
vote
1answer
75 views

Guests leaving Hilbert's Hotel?

I am a layman in this field so my understanding of the problem of "Hilbert's Hotel" is limited to the popular version presented to the public. We know that Hilbert's Hotel can accommodate any finite ...
0
votes
1answer
26 views

Infinite non deviating slope

Okay, I had a question that my math teacher didn't know the answer to, and that I haven't found an answer for on the web. Say you are graphing a system of equations, right, and you have (...
2
votes
0answers
20 views

How to count the number of rebound cycles of a ball in a 1d system after time t, where velocity doubles every rebound [closed]

This is something I dreampt up in a physics lab about imagining infinity but I never got round to modeling it. Seems like it could get out of hand pretty quickly! Imagine we have a ball in a 1d ...
12
votes
4answers
186 views

Prove that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$ [closed]

Using a graphing calculator, it seems that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$. How can this be proven?
2
votes
1answer
31 views

Approximating end behavior of a function by plugging in infinity

In Algebra 2, I learned to be able to tell if the end behavior of a function has an asymptote, approaches infinity, approaches zero, etc, by plugging in numbers closer and closer to infinity, or by ...
0
votes
1answer
38 views

Concept similar to extended real line in higher dimensions?

I have a question related to the notion of extended real line. I am a very beginner of this topic and in what follows I might say things that look no-sense for an expert in the field. The extended ...