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Questions tagged [infinity]

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

0
votes
1answer
43 views

Decomposition of an $\aleph_1$ set into $\aleph_1$ sets $\aleph_1$ [on hold]

Can we justify the claim Any set of cardinality $\aleph_1$ can be expressed as the disjoint union of $\aleph_1$ sets of cardinality $\aleph_1$ in a simple yet reasonably-correct way?
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2answers
50 views

Difference between infinite and undefined [on hold]

In questions, sometimes I see these terms being used interchangeably. But also some lecturers say that they are different. I am pretty confused with these. Please explain.
-1
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0answers
50 views

What is the logic behind $\max\{-\infty,-\infty\}= -\infty$

We are in $\bar{\mathbb{R}}$ therefore both elements are in the set $\bar{\mathbb{R}}$ But how can you compare infinities? Because max is defined as $z=\max{A}\iff z\geq x\in A$.
2
votes
0answers
41 views

How to determine how fast something is going towards infinity?

Just for a little bit of information I'm more of a programmer and less a mathematician so if some of my terms seem out of place it is due to a lack for formal training in Math. While working on my ...
7
votes
7answers
357 views

Error evaluating $ \lim_{x\to 0}\frac{x-\tan x}{x^3} $

Evaluate the limit: $$ \lim_{x\to 0}\frac{x-\tan x}{x^3} $$ I solved it like this, $$ \lim_{x\to 0} \left({1\over x^2} - \frac{\tan x}{x^3}\right) =\lim_{x\to 0}\left({1\over x^2} - {\tan x\over x}...
4
votes
1answer
39 views

Billiards in a holey square

Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because ...
-1
votes
1answer
31 views

Could infinity have a numerical value?

For example, $$\frac{1}{1}=1\quad \frac{1}{2}=0.5\quad \frac{1}{3}=0.\overline3\quad \frac{1}{10}=0.1$$ so the larger the denominator is, the smaller the number is. Would this mean that $\frac{1}{\...
-5
votes
0answers
39 views

Sum of natural numbers? [duplicate]

Is the sum of all natural numbers equal to infinity or -1/12. I have seen convincing evidence on the Internet that the solution is -1/12. But natural numbers are all positive (unless we consider zero),...
-5
votes
2answers
53 views

Why $ \sum_{k=0}^{\infty} \frac{n^k}{k!} = e^n$? [closed]

A lecture note used the following claim: $$ \sum_{k=0}^{\infty} \frac{n^k}{k!} = e^n$$ Why is that?
0
votes
2answers
57 views

Is $\infty$ undefined? [duplicate]

I am confused at $\infty$ in a lot of ways. First, we sometimes say that ${1\over 0}=\infty$. That gives us these confusing calculations.$${1\over0}=\infty$$ $$1=0*\infty$$ $$2=(0*2)*\infty$$ $$2=0*\...
0
votes
2answers
27 views

limit as x approaches infinity from the left (Notation)

This question doesn't involve a specific problem, instead focuses on notation. With limits that approach infinity, is it incorrect to say that the limit approaches infinity from the positive/negative ...
0
votes
1answer
23 views

Approximating when variable to infinity

In a book on algorithms I read that $n^2 (1+\log n)$ as $n$ approaches infinity is approximated to $n^2 \log n$. I am not sure if I understand reasoning in this. Is it because $1+\log n$ grows so ...
0
votes
1answer
30 views

Infinite solids similar to Gabriel's Horn

Do any solids of revolution exist with properties similar to Gabriel's Horn (i.e. a geometric solid with finite volume but infinite surface area)? Please restrict your answers to functions not in the ...
0
votes
1answer
12 views

Is the following statement about matrix norms of matrix products correct?

Let $A$ be a real-valued, square matrix and define its 2-norm as: $$||A||_2 = \sqrt{\max_i\lambda_i(AA^T)}$$ where $\lambda_i(AA^T)$ denotes the $i^{th}$ eigenvalue of the product $AA^T$. Now ...
-1
votes
0answers
25 views

Evaluate $\int_{-\infty}^{\infty}xe^{-x^{2}}dx$ [duplicate]

Evaluate improper integral: $\int_{-\infty}^{\infty}xe^{-x^{2}}dx$
1
vote
1answer
37 views

Integral as a limit of an infinite sum of infinitely narrow rectangles

I tried to approach integrating by filling the space with infinitely many infinitely narrow rectangles. $a$ is the left bound $b$ is the right bound $n$ means the number of rectangles, approaches $\...
-3
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0answers
70 views

I need a help with my research. Just asking for inventive answer to my question. [closed]

Let's take a value and give it to x. Let's say x has its growth doesn't matter how much. Let's describe his growth on a number scale like a man on a boat going in a direction of a river flow. Let's ...
4
votes
2answers
119 views

Why can’t you reassign the ‘mystery number’ in Cantor’s diagonal argument to a new number in the natural numbers?

I don’t want to claim that I have ‘refuted Cantor’ or something here, I just want to understand it adequately. I do understand that the proof works something like this: You assume that you can map ...
2
votes
1answer
828 views

What is the status of the Axiom of limitation of size? (adrift for almost a century now)

On reviewing the wiki article Axiom of limitation of size I come away with the impression that there are issues surrounding this 'maybe too powerful' principle/heuristic/doctrine/axiom that haven't ...
0
votes
1answer
30 views

Probability for an infinite set

The way probability is defined as the expected value works for finite sets. The probability of getting heads is out of two possible outcomes, heads or tails. If we asked the probability out of an ...
0
votes
0answers
31 views

The expected value of a random function

Suppose we have a 'random function' $f(x)$ that will be one of $n$ functions $f_1(x), ..., f_n(x)$ with probabilities $p_1, ..., p_n$. Intuitively, it might seem sensible to define $$E[f(x)] = \sum_{...
0
votes
2answers
85 views

Example of a set of real numbers that is Dedekind-finite but not finite

Without assuming $AC$, can we find an explicit example of a subset of $\mathbb{R}$ such that it is not finite but it is Dedekind-finite?
0
votes
2answers
67 views

Proving $\lim_{x\to-\infty}x^{-k} = 0$

Please prove $\lim_{x\to-\infty}x^{-k} = 0$ for k a natural number using the definition of this limit. I am having problems discovering the proof.
3
votes
2answers
330 views

Can I conclude that $\lim_{x\to0^+}\frac{x^2}{e^{-\frac{1}{x^2}}\cos(\frac{1}{x^2})^2}$ is infinite or it doesn't exists?

$$\lim_{x\to0^+}\frac{x^2}{e^{-\frac{1}{x^2}}\cos(\frac{1}{x^2})^2}$$ My intuition is that the denominator goes to 0 faster and everything is non-negative, so the limit is positive infinity. I cant ...
0
votes
3answers
65 views

How to solve the limit $\lim\limits_{x\to \infty} (x \arctan x - \frac{x\pi}{2})$

Next week I have a math exam. While I was doing some exercises I came across this interesting limit: $\lim\limits_{x\to \infty} (x \arctan x - \frac{x\pi}{2})$ After struggling a lot, I decided to ...
1
vote
0answers
47 views

A ray and a line intersect, and the usefulness of geometric, euclidian beyond infinity?

A 2D euclidian plane R2 contains a line and a ray. The line is vertical and positioned where $x = 0$. The point on the line where $y = 0$ is labelled $m$. One end of the ray is at $(x, y) = (1, 0)$ ...
2
votes
2answers
59 views

Does the constant $C$ in this solution to a differential equation equal infinity?

The problem is $y' = -\frac{1}{t^2} - \frac{1}{t}y + y^2;\ y_p = \frac{1}{t}$. My solution is $$\begin{align} y = \frac{1}{t} + B &\implies y' = -\frac{1}{t^2} + B' \\ &\implies -\frac{1}{t^...
1
vote
5answers
601 views

If irrational numbers are uncountable, then why did I find this? [closed]

I understand that irrational numbers are uncountable. I've seen the proof and it makes perfect sense. However, I came up with this (most likely false) proof that says that they're countable. ...
0
votes
3answers
51 views

Expected value of a random positive integer

If I theoretically had a true random number that picks any positive integer, then the expected value should be infinity by the following logic. Expected value will be n/2 for the sequence $ 0 ,1,2,3,.....
1
vote
3answers
74 views

Sum of Infinity of Trigo to Pi

I am currently working on a proof with a good friend of mine that involves adding more and more triangles to the sides of a regular polygon but keeping the longest diagonal constant until eventually, ...
0
votes
1answer
30 views

Confusion arising from the 'infiniteness' of a sequence.

Let $ X = \{ i | i \in \mathbb{N} \} = \mathbb{N}$. Then is there an infinite sequence $ \{t_i \} $ with terms from $X$ such that $\forall i \ t_i > t_{i+1} $ viz. a strictly decreasing sequence. ...
0
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0answers
8 views

Counting equivalence classes. [duplicate]

Let $X$ denote the set of real transcendental numbers. Define the relation $\sim$ on $X$ by $x\sim y $ iff $x-y \in \mathbb{Q}$. Let $Y$ denote the set of equivalence classes generated by $\sim$ ...
4
votes
1answer
35 views

Infinite set of positive integers - choose infinitely many to be relative primes or not

Given a set of infinitely many positive integers. Is it always possible to find a subset of this set which contains infinitely many numbers such that any two numbers in this subset are relative primes ...
0
votes
1answer
58 views

Parity of an infinite exponential function (What shape is $y=x^\infty$)

When you have functions of the form $x^c$, the shape of the graph is symmetric with even integers for c (└┘-shape) and for odd integers it is non-symmetric (┌┘-shape) When dealing with limits as c ...
1
vote
2answers
85 views

Sum to Infinity of Trigonometry to $\pi$

For $$y=\sum_{n=0}^a2\cdot2^n\cdot\tan\left(\frac{45}{2^n}\right)\cdot\sin\left(\frac{90}{2^n}\right)^2$$ I am currently working on a proof with a good friend of mine that involves adding more and ...
0
votes
1answer
212 views

If the law says “Provide a valid solution for 0x = 50, or go to jail”, can we avoid being jailed? [closed]

Law Stack Exchange has a question about a hypothetical law that everyone is meant to be guilty of breaking. What if a law is literally impossible to follow? While that's an interesting idea, they ...
1
vote
2answers
48 views

Using ∞ as an algebraic term

Would it ever be possible to use ∞ as an algebraic term, putting it in equitations with non infinite terms, incorporating a coefficient? Would a new type of infinity need to be defined or could aleph ...
6
votes
2answers
168 views

solutions to $\int_{-\infty}^\infty \frac{1}{x^n+1}dx$ for even $n$

I was playing around with glasser's master theorem and integrals of the form $$\int_{-\infty}^\infty \frac{1}{x^n+1}dx$$ I observed that for positive, even values of n, the solution to the integral ...
1
vote
4answers
72 views

Limit $\lim_{x\to\infty}\left(1-\frac{a^2}{x^2}\right)^{x^2}$

I found this example in a textbook, and I understand the author's reasoning and I also reached the same answer using L’Hôpital’s rule. However, I have two issues: Firstly: For any finite $a$, then ...
1
vote
1answer
29 views

Can I prove that a 2-variable limit does not exists if the limit on a curve is infinity?

Consider a 2 variable function $f(x,y)$ and the limit $$\lim_{(x,y)\to (0,0)} f(x,y)$$ If I find two continuous functions $\gamma_1(t)$ and $\gamma_2(t)$ such that $\gamma_1(0)=\gamma_2(0)=(0,0)$ ...
3
votes
2answers
167 views

Could one argue that $10 \cdot 10 \cdot 10 \cdot 10 \cdots$ is equal to 0?

The thing about this is, if we assign a variable: $$x=10 \cdot 10 \cdot 10 \cdot 10 \cdots$$ and then enclose all but one multiplicand in brackets (as multiplication is associative) and we get: $$x=10 ...
4
votes
1answer
39 views

Is the concept of dimension still well defined for non-finite dimensional spaces? [duplicate]

The question is quite simple: if $\mathbb{V}$ is a vector space and $B$ and $B'$ are basis for $\mathbb V$, then do $B$ and $B'$ have the same cardinality? I've tried to answer the question as ...
0
votes
0answers
44 views

Can we state the existence of infinite set without infinity axiom? [duplicate]

I have a question about infinity axiom in ZF and maybe, it has nonsense. So I apologize in advance if it is the case. In ZF, the infiny axiom can be state as $\exists X(X\neq\emptyset\wedge\forall x\...
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votes
3answers
92 views

Evaluating the limit $\lim_{x \to \infty}\left(\sqrt{x^4 -x^3+1}-\sqrt{x^4+15x^2-5}\,\right)$

I wanna know how to do this limit $\lim_{x \to \infty}\left(\sqrt{x^4 -x^3+1}-\sqrt{x^4+15x^2-5}\,\right)$
4
votes
2answers
75 views

Is there a set in which division of 0 by 0 is defined?

The reason I ask this is that I've discovered that, even though they don't satisfy all field axioms, there are sets called projectively extended real number line and Riemann sphere, which are ℝ∪{∞} ...
1
vote
2answers
83 views

How would the set $\mathbb{N}= \{1, 2, 3, \dots \}$ be a finite set according to Definition 1.3.1 from Bartle's *Introduction to Real Analysis*?

This is from Sherbert and Bartle's Introduction to Real Analysis. 1.3.1 Definition (a) The empty set is said to have $0$ elements. (b) If $n\in \mathbb{N}$, a set $S$ is said to have $n$ elements ...
0
votes
3answers
363 views

Sum of all natural numbers.

Okay, I do know that there are three ways of showing that it equals $-1\over 12$: 1) The Reimann zeta function calculated for $-1$(see picture) 2) The one involving Grandi's series, the series $1-2+...
2
votes
1answer
25 views

Definitions of normalisers for infinite groups

If G is a group and A is a subset of G, the normaliser of A in G can be defined as either (1) $N_G(A) = \{g \in G\ |\ gag^{-1} \in A, \forall a \in A\}$ (2) $N_G(A) = \{g \in G\ |\ gAg^{-1} = A \}$ ...
0
votes
2answers
77 views

$(I-A)^{-1}=\sum_{i=0}^\infty A^i$

Let $V$ be a finite dimentional normed vector space and let $A$ a linear transformation from $V$ to $V$ such that $\left \| A \right \|<1$. Show that the linear tranformation $I-A$ is invertible ...
0
votes
0answers
24 views

Eigenvalues of an infinite matrix

I have matrix elements $$ H_{mn} = \begin{cases} \frac{1}{(m-n)^2} & m \neq n \\ 0 & m=n \end{cases}$$ If the indices extend to infinity what are the eigenvalues of this matrix? I'm sure ...