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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434 views

“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
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152 views

Is it a “paradox”, or a flaw in the question?

(Clearly not a pardox per-se but I would like to hear what you think) The basic riddle (not a very interesting one even) goes as follows: A first client comes into a barber shop, takes a hair cut ...
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215 views

Quantifying infinitely large sums such as $\sum_{x\in\mathbb{R}^+} x$

I thought of this as a student in calculus years ago, and it may be a silly kind of question. I wondered if there were notions of different sizes of infinity a series might sum to, which then lead me ...
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36 views

The behavior of the 3D wave equation close to the origin

The general solution to the three dimensional wave equation is \begin{equation}u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r} \end{equation} where $F$ and $G$ are arbitrary functions. I want to ...
3
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43 views

Kempner's series vs harmonic series (convergent vs divergent series via exclusion)

$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots$$ Okay so we all know the harmonic series is divergent right? But apparently when you remove all the terms that has a nine in it, ...
3
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73 views

How does $\ln(x)$ blow up at $0$ and $\infty$.

In general: How do I figure out how fast a function blows up at a certain point or infinity? How fast does $\ln x$ blow up at $0$? Does it blow up as fast as $1/x$, $1/x^2$, or maybe faster than any ...
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100 views

Infinity Gradient

I calculate infinity gradient, but I am not sure is this correct.
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205 views

Limits of infinite processes that terminate in finite time - checking my understanding?

I am a computer scientist by training, but have a fair amount of math background that I've picked up through classes, teaching, and general interest. A student of mine posed a question to me. I think ...
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60 views

Cantor, longish lines and the Landau -o notations

in general terms this question is about the behaviour of functions of a real variable as their argument $\rightarrow \infty$. i will present the matter as concisely as i can, but my presentation will ...
2
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25 views

Prove the squared vector 2-norm is $\leq$ sum of 1-norm and infinity-norm

How do I prove that $$\|x\|_2^2 \leq \|x\|_1 \|x\|_\infty?$$
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52 views

For Vector Spaces V and W with one Infinite Dimensional , is Hom(V, W) Isomorphic to Hom(W, V)?

If V and W are both finite then clearly Dim (Hom(V, W)) = Dim(V).Dim(W) = Dim(Hom(W, V)) so they are isomorphic. I'm not so sure if one is infinite. An "infinite matrix" construction for a linear ...
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94 views

mathematical limit for a ouroboros torus

The other day i was watching an episode of Tom and Jerry in which a similar situation was present toms head comes out of his own mouth. My head hurts when i think how is that even possible so i ...
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36 views

Countability of unions versus products

Let $D_{n}$ be a set with $2^{n}$ elements for $n=1,2,...$. Let $A = \bigcup_{n=1}^{\infty}D_{n}$, and let $B = \prod_{n=1}^{\infty}\{0,1\}$. Let $A_{k} = \bigcup_{n=1}^{k} D_{n}$, and let $B_{k} = ...
2
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191 views

Fine-grained way to measure infinity

It is known that the cardinality of $R$ is equal to the cardinality of $R^2$, $R^3$, etc. But, intuitively these sets have different sizes. A possible way to formalize this intuition is to talk about ...
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35 views

Friend B and C have eaten zero apples. How many more apples has C eaten?

Friend $A$ claims that he has eaten $1$ apple today. Friend $B$ responds. Congrats, I have eaten $0$ apples, so that is $\infty$ more apples than me. Friend $C$ says, but I have also eaten $0$ ...
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31 views

Complex residue at infinity of $f(z)=\frac{z^5}{\sin\left(\frac{1}{z^2}\right)}$

I'm having trouble finding residue of the function $$f(z)=\frac{z^5}{\sin\left(\frac{1}{\large{z^2}}\right)}$$ at infinity. Wolfram kindly informs that it is equal to $-\frac{7}{360}$ (and gives ...
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22 views

Is it formally correct to take the limit of infinitely many terms?

Let us take the most basic example: $\lim_{n\to\infty} \frac{n!}{n^n}$ $\frac{n!}{n^n} = \frac{n(n-1)...1}{n \cdot n...n} = \frac{n}{n} \cdot \frac{n-1}{n} \cdot [...] \cdot \frac{1}{n} \leq ...
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32 views

Probability of termination of random teleportation

In Minecraft, with mods, there's a liquid called Resonant Ender, which if you touch it, teleports you randomly up to 8 blocks on both the north-south and east-west axes. Consider an infinite sea of ...
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35 views

Show that $l_{2}(J)$ is Hilbert Space for Countably Infinite Set?

The inner product is \begin{equation*} \langle u, v \rangle = \sum\limits_{j \in J} u_{j} \overline{v_{j}} \end{equation*} where $u,v$ are vectors and $J$ is the countably infinite set $J = ...
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82 views

Is there a change-of-variables solution for integrals from negative infinity to a constant?

I found a fantastic and generalizable substitution technique for computing definite integrals that go to infinity from either negative infinity or a constant, regardless of the function (sorry for the ...
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64 views

How does one prove that two sequences are equal at infinity?

I came up with $e = \sum_{n=0}^\infty \frac 1 {n!}$ (see here) I am now trying to prove that this is equivalent to $\lim_{n\to \infty} {(1+\frac1 n)}^n$ In general, how would one go about such a ...
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32 views

How do I show that infinite application of this function gives a constant?

I want to show that g(x) returns the same value independent of x and hence is a constant. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
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100 views

Is it true that the slope of a vertical line times the slope of a horizontal like don't equal $-1$, even though they're perpendicular?

I know that the slopes of two lines that are perpendicular have a value of $-1$ when multiplied because they're opposite reciprocals (e.g. $5$ and $-{1\over 5}$), but what if there's a horizontal and ...
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156 views

Hilbert's hotel with uncountably infinite rooms: can you fit $\mathbb R^2$ guests?

I'm trying to expand on Hilbert's paradox. The original version states that: Suppose there is a hotel with a countable infinity of rooms (eg. $\mathbb N$), all of which are occupied. ...
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87 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
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145 views

What does “radical cube zero” mean?

My hobby is taking comics way too seriously. And I just came across a math topic. In a certain comic (Fantastic Four 51, according to some polls the greatest comic issue ever) there's a machine for ...
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111 views

Limit of constant function

I was reading a proof using Markov Chains in a finite state space $E$. Denote $p_{ij}(n) = P(X_n = j | X_0 = i)$. Since the state space is finite, then probability of landing somewhere in the state ...
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56 views

Prove infinity arithmatics

How do you prove $ \infty * (-\infty) = -\infty$ or $ \infty +\infty = \infty$? I thought it is an axiom, but have been there's is proof for that.
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63 views

Meaningful measures for comparing infinite dimensional geometric objects

I have two infinite-dimensional convex polytopes, call them $A$ and $B$. I know that $B$ is completely contained within $A$, and I want to say something meaningful about their relative sizes. From ...
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165 views

Relationship between ordinals and rank of well founded relations on $\mathbb N$

I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals. Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
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125 views

maximize an objective function with an infinite component

Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
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114 views

Hadamard regularization isn't working out

As part of an exercise in a grad course on "mathematical methods" (always such a helpful name), I've been asked to evaluate $I=\int_0^{1/2}{(x^2-x+c)^{-2}dx}$ as a Hadamard finite part integral for $0 ...
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57 views

Probability in infinitary logic

Let X be a random variable taking the value 0.2 with probability 0.2, 0.4 with probability 0.4 and 0.8 with probability 0.2 and 1.0 with probability 0.2. Using Infinitary logic I can ask the ...
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55 views

If we think of infinity as a number, how does it affect the compactness/completeness of a metric space?

I was recently reviewing some notes regarding compactness, in which the sequential definition is given i.e. "$A$ is compact if any sequence in $A$ has a subsequence which converges to a limit in $A$. ...
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32 views

If I can prove f(n) = g(n+1) by induction when n is finite, Can I prove f(n) = g(n) by taking n = $\infty$

I have to prove f(n) = g(n) when $n = \infty$. Now I can prove f(n) = g(n+1) by induction when n is finite. Can I say $f(n) = g(n)$ by taking $n = \infty$?
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23 views

What is the origin of the distinction between assignable and inassignable number?

Leibniz described his infinitesimals as being inassignable numbers in a number of texts, e.g., in his Cum Produisset that was analyzed in detail by H. Bos in a seminal text dating from the 1970s. The ...
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37 views

integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz} $ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
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30 views

How to solve a particular indeterminate form

So the answer says $$\lim_{x\to \infty}x^2\sin\left(\frac1x\right)=\lim_{h\to 0^+}\frac1h\frac{\sin h}h$$ How does the transformation work?
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36 views

Definition of the rank of infinite matrix

How is the rank of an infinite matrix defined? Is it the same as in the finite case, i.e. the number of elements in a basis for some matrix? How are the dimensionalities of the column and null ...
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33 views

$\Sigma^{\infty}_{n=1}(-1)^n[ \frac{\pi sin(n\theta)}{n}-\frac{2cos(n\theta)}{n^2}]=0$

Is $\Sigma^{\infty}_{n=1}(-1)^n[ \frac{\pi sin(n\theta)}{n}-\frac{2cos(n\theta)}{n^2}]=0$ true? This is came out from my Fourier series computation, according to the answer, that sum should be zero ...
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62 views

Probability of Unions to infinity

I had a question about the probability of unions to infinity. 1) Everyone in a group of $N > 3$ people writes their name on a slip of paper and drops the slips into an urn. Then, one at a time ...
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22 views

Probability of getting black or white

If I have a bag of black and white marbles, say 10 black and 10 white, I have a 50% chance of either getting black or white. What if the bag were infinite? Can one talk about probability when ...
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41 views

Good papers on Cantor's infinities?

I'm trying to get a mathematical treatment on Cantor's infinities, but can't seem to find papers merely by googling. Any recommendations?
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61 views

Laplace transform and “imaginary infinity”

I was recently studying Laplace transform for the first time, and I'd like to ask the following thing: there was an integral with limit of integration, something like that: a+j×infinity, j the ...
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44 views

find the supremum

Hi I'm trying to figure out for which values of $w$ of $u(x,t)$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is ...
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22 views

Compute a sum with finite and infinite elements

I would like to compute the following summation: $$ s = \sum_{i=1}^n a_i \, \Phi^{-1}(u_i) $$ where $\Phi^{-1}$ is the inverse of the standard Gaussian distribution function, $a_i$ are some real ...
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41 views

Limit of a sequence question

I have the following question in my assignment which I couldn't solve: Let $({a_n})$ and $({b_n})$ be two sequences, such that $\lim\limits_{n \to \infty}({a_n}{b_n}) =0 $ I have to prove if the ...
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28 views
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42 views

Mapping from 'one infinity to another'.

I've been wondering about this for a while now. Consider the function $f(x)=\frac{1}{x}$. When $x\in(0,1)$, the function maps to the interval $(1,\infty)$. Conversely, anything in the interval ...
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26 views

Convention on infinity comparision

What's the difference between $-\infty \leq a \leq \infty$ and $-\infty < a < \infty$ conceptually or otherwise. It doesn't really affect the solution to the solution of this problem I'm ...