Somewhere beyond the numbers lies the concept of Infinity. But what exactly does "infinity" mean? What rules does it obey? What interesting properties does it have?

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374 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
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69 views

The meaning of infinite series $\sum_{i=0}^\infty 2^{-i}$, its relation to partial sums and Cantor's diagonal argument

Let's define $S(n)$ as $S(n) = \sum_{i=0}^n 2^{-i}$. Obviously, $\lim_{n \to \infty} S(n) = 2$ and also $\forall n \in \mathbb{N}, S(n)<2$. Now my questions are about $Q = \sum_{i=0}^\infty ...
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70 views

Regarding playing an infinite number of games that could last infinitely long amounts of time

So after watching the last Stanley cup game, a problem popped up in my head for which I have no solution. Say we have a game, like a hockey game, that has the possibility of going on forever. Of ...
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1answer
92 views

Assigning values to divergent integrals

I'm interested in the (obviously divergent) integral $$ \int_{-\infty}^\infty dx e^{-x f}\ ,$$ where $f$ is real. Is there any way to meaningfully assign a value to this integral? I was thinking of ...
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1answer
37 views

How do we evaluate this limit?

$$ A_N(x) = \lim_{N\to \infty}(\sin(x)/x)^N $$ The solution to this problem is given as, $$ A_N(x) = \exp( -Nx^2/6). $$ The problem is solved through Taylor series expansion for $\sin(x)$. And ...
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344 views

What is infinity in complex plane and what are operation with infinity extended to complex numbers?

For a real number $a$, $$\infty + a = \infty,$$ and if $a$ is positive, $$\infty \cdot a = \infty$$ What is $\infty + a$ and $\infty*a$ if $a$ is non-zero complex number, where $\infty$ is real ...
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36 views

Is the number of points on a plane larger than the number of points on a line?

The number of points on a line is uncountably infinite. The number of lines on a plane is uncountably infinite. It seems like it follows that there would be an uncountably infinite number of points on ...
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22 views

limits as $x\rightarrow\pm\infty$ of indeterminate forms $\frac{a^x+b^x}{c^x+d^x}$, where $a,b,c,d\in\mathbb{R}$

Good day sirs would you kindly help me to find the limit of $\frac{a^x+b^x}{c^x+d^x}$ as $x\rightarrow\pm\infty$, where $a$,$b$,$c$ and $d$ are real numbers? I already know how to use the L' ...
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42 views

Number of paths in a graph with infinite nodes

Does a graph with infinite nodes that is not fully connected have a countably infinite or a uncountably infinite number of paths originating from a single node? We are only concerned with paths that ...
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389 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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112 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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62 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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68 views

Suppose some theory T has countably many axioms, how many models of $T$ are there of cardinality $\aleph_1$,$\aleph_2$,$\aleph_{\omega_1}$?

Setting Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes. So we see $T$ has ...
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187 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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56 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 ...
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56 views

Infinity Gradient

I calculate infinity gradient, but I am not sure is this correct.
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46 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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48 views

Is this set countably infinite or not?

"Far away, in the heavenly abode of the great god Indra, there is a wonderful net that has been hung by some artificer in such a manner that it stretches out infinitely in all directions. In ...
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35 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} = ...
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41 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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31 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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26 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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36 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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58 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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74 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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135 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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97 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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58 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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134 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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119 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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101 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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52 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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38 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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21 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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22 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 ...
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23 views

is it in fact impossible to construct a machine which can know if a macine ever prints a character?

In $\S\ 8$ of his paper "On computable numbers, with an application to the Entscheidungsproblem" Turing uses his proof that $\mathfrak{D}$ (a machine which given the S.D. of another machine ...
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15 views

Integration help with Hermite polynomials or direct integration!

This is my formular: $$ \psi_2=N_2 (4y^2-1) e^{-y^2/2}, $$ where $y=x/a$, $a= \left( \frac{\hbar}{mk} \right)$, $N_2 = \sqrt{\frac{1}{8a\sqrt{\pi}}}$. Here is my integral: $$ <x^2> = ...
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37 views

Estimating the mean Euclidean distance between two overlapping, not-matching shapes

I’d like to determine the mean distance between two irregular overlapped, not-matching shapes ($X$ and $Y$). In $Figure 1$, $X$ is “visually above” $Y$, and that’s why we can’t see part of the $Y$ ...
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52 views

Circles and the continuum hypothesis

I was trying to understand the undecidable nature of the continuum hypothesis and came up with the following question: The set of circles with a rational diameter is countably infinite (with ...
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41 views

Check proof of union of denumerable sets is denumerable too

I need to prove: If $A$ and $B$ are denumerable sets then so is their union $A\cup B$. In this case, denumerable is defined as: A set $X$ is said to be denumerable if there is a bijection ...
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47 views

Cardinality of infinite between the set of rationals and set of reals

I remember learning that whether or not there is a cardinality of infinity between the set of rational numbers and the set of real numbers is unprovable. Is this true, and if so, how do we know it to ...
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44 views

How do you call a scale that starts at $∞$, has $1/n$ divisions and tends to $0$?

A linear scale $2n$ divisions: 0 2 4 6 8 Logarithmic scales $10^n$ divisions: 1 10 100 1000 10000 ...
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Dividing an infinite plane into regions

I am currently working on a computer program for computing layout of graph-based diagrams. Their content is placed in an "infinite" 2D plane with cartesian coordinates in the center of the diagram. ...
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19 views

Is it possible to extend the complex plane affinely and projectively at the same time?

Is it possible to extend the complex plane affinely and projectively at the same time? That is by adding both the positive infinity (with based on it directed infinity) AND the unsigned complex ...
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44 views

Help with calculating infinite sum

I'm working on a problem, and I'm stuck in the calculations of finding $\sum_{0}^{\infty}\frac{1}{1+n^2}$ Suggestions on how to approach this calculation? Thanks! (Also, I used Fourier to get to ...
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112 views

How to observe infinity?

In my calculus course, there's example stated on the book: Given that $M$ is an ordered set and the sequence $\{a_n\}\subset M$, prove that there's a (weakly) monotonic subsequence of $\{a_n\}$. ...
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164 views

Optimization of infinite series. Dynamic programming.

I wonder how to optimize an infinite series using Classical (Lagrangian) method. For example:$$\sum_{t=1}^{\infty}\beta^{t-1}\ln c_t$$ subject to the constraint $$c_t+b_t\leq y_t+(1-r)b_{t-1}$$ ...
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How many different random numbers can you generate between zero and infinity?

My theory is that you can't generate any random number between zero and infinity. But thats kind of unexpected because you can generate 100 different random numbers between 0 and 100 and 1000 ...