# Tagged Questions

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

459 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 ...
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 ...
162 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 ...
217 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 ...
38 views

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

The general solution to the three dimensional wave equation is $$u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r}$$ where $F$ and $G$ are arbitrary functions. I want to ...
50 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, ...
83 views

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 ...
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 ...
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 ...
57 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) ...
39 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 $0$ ...
40 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 ...
78 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 ...
23 views

141 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 ...
80 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 ...
41 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. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
117 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 ...
95 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 ...
148 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 ...
114 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 ...
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.
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 ...
176 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}, ...
127 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 ...
117 views

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 $y\to\infty$...
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 ...
17 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) ...
3 views

63 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 $A$. ...
37 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 $n = \infty$?
24 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 ...
32 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?
### $\Sigma^{\infty}_{n=1}(-1)^n[ \frac{\pi sin(n\theta)}{n}-\frac{2cos(n\theta)}{n^2}]=0$
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 ...
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 (...