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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Proving that if a set $A$ is infinite then necessarily $|A|\geq|\mathbb{N}|$ [duplicate]

A set $A$ is set to be infinite if it is not finite, i.e. if there exists no $n\in\mathbb{N}$ such that $|A|=n$, meaning there exists a bijection $A\leftrightarrow\{1,\dotsc,n\}$. How do I prove that ...
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Infinity Gradient

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

Question about $\infty\cdot0$ as not defined number

Let's say we have this: $$ \infty \cdot 0 + \infty$$ since $\infty\cdot0$ is not defined can we do this: $$ \infty \cdot(0 + 1) = \infty \cdot 1 = \infty $$ and therefore can I say $\infty\cdot0 + ...
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Limit question $\infty^{0}$ type

$$\lim_{x\to\frac{\pi}{2}^-} (\tan x)^{\cos x}$$ I just tried to write $e^{\ln(\tan x^{\cos x})}$ form but I couldn't solve the limit.
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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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96 views

Something I don't understand about Hilbert's grand hotel

So I want to know if Hilbert's hotel "story" holds for this statement: $\wp (\mathbb{N}) \sim \wp (\mathbb{N})\smallsetminus \left \lbrace\emptyset\right\rbrace$ So, If the statement wasn't talking ...
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What is the difference between $\omega$ and $\aleph_0$?

The book I'm using says that the cardinality of a set $X$ is the least ordinal $\alpha$ such that $|X| = |\alpha|$. So then $\omega = \aleph_0$, but $\omega + \omega \ne \omega$, while $\aleph_0 + ...
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75 views

Limit of $\frac{n!}{(n+1)!}$ as n approaches infinity.

I know that factorials grow faster than any exponential function, but what if you put two factorials up against each other? My problem is finding the limit of: $$\frac{n!}{(n+1)!}$$ as $n$ ...
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69 views

Uniqueness Proof for solution to $\nabla^2 G(\textbf{r}) = \delta(\textbf{r})$ with $G \rightarrow 0$ when $|\textbf{r}| \rightarrow \infty$

I'm having difficulty understanding the derivation of solution to this equation: $\nabla^2 G (\textbf{r}) = \delta(\textbf{r})$ with $G \rightarrow 0$ when $|\textbf{r}| -> \infty$ in $R^n$ where ...
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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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115 views

Irrational numbers in between $n$ and $n+1$

Is the amount of irrationals numbers in between consecutive integers always the same? is this amount infinite?
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52 views

About the infinite geometrical sequence factored with n [duplicate]

I just came across this thread, and i asked myself: I know that $\sum^\infty_{n=0} x^n = \frac{1}{1-x}$ But what happens when we set up the sum like $$\sum^\infty_{n=0} nx^n = ?$$ There is ...
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74 views

Question about probability in infinite game of chance

Imagine a game where you start with \$100 and toss a coin repeatedly. If it's heads - you lose \$1, if its tails - you double your money. Game ends when you lose all the money. Given infinite amount ...
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Proof that the harmonic series is < $\infty$ for a special set..

In one of my books i found a very interesting task, i am really curios about the solution: Let $M = \{2,3,4,5,6,7,8,9,20,22,...\} \subseteq \mathbb{N}$ be a set that contains all natural numbers, ...
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63 views

Divergent to $\infty \Rightarrow$ Divergent?

In our lecture, we defined a sequence $\left(a_n\right)_{n\in\mathbb N}$ to be divergent if it does not converge, and additionally to be divergent to $\pm \infty$, iff: $$\forall \epsilon \in \mathbb ...
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Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality ...
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51 views

Finding the sum and nth term of a series

How do you find the value of this series? $$\sum^\infty_{n=2}{\frac{2^n + (-1)^n}{4^n}}$$ I tried writing out the series at $n=2, n=3,$ and $n=4$, and I attempted to look for a pattern with which ...
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110 views

Will the Declaration of Independence ever show up in pi? [duplicate]

If pi goes on forever and is completely random, if ascii would be mapped onto pi would you eventually find the Declaration of Independence in it? If so, by what digit of pi can we reasonably expect ...
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55 views

Hypercomputation & Higher Dimensional Variants of Conway's Game of Life

Conway's Game of Life is a simple and important mathematical game with some rules of evolution in a two dimensional space. It appears in many subjects in mathematics, artificial intelligence and ...
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169 views

Why does Infinity x Zero not Equal One? [duplicate]

Why does Zero Times Infinity not equal One ($0 \times \infty \neq 1$)? If Infinity = $\infty$ and Zero = $\frac{1}{\infty}$ Then Zero Times Infinity = $0 \times \infty = \frac{1}{\infty} \times ...
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Example of set of cardinality $\aleph_2$

I am looking for an example of a set of cardinality $\aleph_2$, such as the continuum is an example for cardinality $\aleph_1$.
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How to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$?

I need to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$.
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Why aren't there $+\infty^{+\infty}$ real numbers?

I was reading this pop math piece on "the different sizes of Infinity." The article explains why the real numbers are uncountably infinite. Taking a real number, my uneducated mathematical mind ...
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Can I say that a fixed constant is less or equal infinity?

Mathematically speaking, given $c\in\mathbb{R}$, can I say that: $c\leq\infty$? E.g., is $10 \leq \infty$ a correct mathematical statement? I know this comparison is true in computer arithmetic, ...
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Difference in treatment of Infinity and Undefined

I understand that $$1)\; \lim_{x\to0}\frac1{x} = +\infty$$ $$2)\; \frac1{0} is\,undefined $$ If both infinity and undefined ...
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Zero and infinity

Introduction [can be skipped without loss of generality]. This question was closed and, recently, deleted, perhaps for good reason. It did have an answer with 10 upvotes, and another (mine) with 15 ...
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1/∞ is 0 or infinitesimal?

Since ∞>0 , so 1/∞>0, thus I think 1/∞ should be infinitesimal, but the calculus book says $\displaystyle \lim_{x \to \infty} \frac{1}{x}= 0$ So is 1/∞ 0 or infinitesimal ? P.S.I mean 1/∞ and ...
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Was there anybody before Cantor who conjectured existence of infinities of different sizes?

Georg Cantor is formally known as the first one who discovered existence of infinities of different sizes. But the history of thinking about the concept of "infinity" in maths and philosophy goes back ...
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Dividing a number into infinite pieces

Last day in physics teacher said that any number divided into infinitely many pieces is zero.It got me thinking in kind of weird direction so here is what I was thinking about and how I tried to ...
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Calculate $\sum_{n=1}^{\infty}(\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2n+4})$

I am trying to calculate the following series: $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)}$$ and I managed to reduce it to this term ...
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Are there smaller orders (cardinalities) of infinity?

I am using this source as a basis for the language to ask this question. Considering the topic of degrees of infinity, are there smaller degrees than ℵ0 (aleph null, also called ω)? ...
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If there's only two infinities, why isn't Calculus affected?

I've been told by a friend that there are (thought to be) only two infinities: the real infinity and the integer infinity. If that's the case, why is $\displaystyle\lim_{x\to\infty}{x \over x^2} = ...
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Inverse of an infinitely large matrix?

This is probably a trivial problem for some people, but I've spent quite some time on it: What is the inverse of the infinite matrix $$ \left[\begin{matrix} 0^0 & 0^1 & 0^2 & 0^3 & ...
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If $\omega + 1 = \omega$, find $\omega$ ($\omega \not= - \infty$ or $\infty$)

If $\omega + 1 = \omega$, find $\omega$ ($\omega \not= - \infty$ or $\infty$). It does not have to be a real number. My teacher gave us this question just to play around with, and my first ...
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A question about infinitie series and pi

This is the sequence that can be used to find an exact value of pi 4/1−4/3+4/5−4/7+4/9−4/11…..(to infinity) = 𝜋 Or (1/1−1/3+1/5−1/7+1/9−1/11….. (to infinity) )= 𝜋/4 Given that we have this ...
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Why can't consecutive irrational numbers be treated mathematically as limits?

I'm a relative newcomer to these stackexchange websites, and this post will serve as my introduction to the Mathematics stackexchange site. After perusing some of the related questions, I found these ...
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Arithemetic series addition

Lets say I have M= 1+2+3+4+5+6+7.... (to infinity) and I have another sequence,N= 6+14+22+30..... (to infinity) is it possible to say that N = 4M +2 ? Or is there another way that I can write ...
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Proof of Nesbitt's Inequality?

I just thought of this proof but I can't seem to get it to work. Let $a,b,c>0$, prove that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$$ Proof: Since the inequality is homogeneous, ...
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Why do we care about the 'rapidness' for convergence?

It is those puzzeling improper integrals that I can't get my head around.... Does the (improper) integral $\frac 1{x^2}$ from 1 to $\infty$ coverges because it is converging "fast" or because it has ...
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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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infinite limit question from Calc I

Find the limit $$\lim_{x\to\infty}\sqrt{x^2+x+1}-x$$ This limit is part of a question involving squeeze theorum, the limit is $\frac12$ but i don't know how to prove it because of the polynomial in ...
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L'Hôpital's as $x$ tends to infinity

I'm searching for the explanation to the limit of: $$ \lim\limits_{x\to\infty} x\, \ln\frac{x+1}{x-1}. $$ I know the answer is 2, but I can't seem to get there. The problem is in my textbook under a ...
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Convergence of this alternating series: $\sum_{k=0}^\infty \frac{(-1)^k}{(k+1)C^k} = C \log \frac{C+1}{C}$

I "heard" the following formula for any $C \ge 1$: $\sum\limits_{k=0}^\infty \dfrac{(-1)^k}{(k+1)C^k} = C \log \dfrac{C+1}{C}$ Is it correct? What would be a proof?
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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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275 views

Did I construct an infinite set equal to $\{1\}$?

Okay, I'm trying to understand the argument that NJ Wildberger gives in the following video: https://www.youtube.com/watch?v=5CiiGdaYEPU He tries to explain why he things infinite sets don't make ...
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26 views

Question about $\lim_{x\to \infty}\frac{\cos(3x)}{e^{8x}}$

$\lim_{x\to \infty}\dfrac{\cos(3x)}{e^{8x}}$ The answer is $0$. Why is the answer $0$? The top oscillates between $-1$ and $1$ and the bottom becomes huge, but since the top is oscillating, ...
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The limit as x approaches infinity

$$\lim_{x\to\infty}x\left(1-\sqrt{1+\frac1{2x}}\right)$$ Can anyone explain how to get this?
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407 views

Limit to Infinity question?

$$\lim_{x\to\infty}\left(-\sqrt{-2x+x^2}+\sqrt{2x+x^2}\right)=2$$ I'm not sure how to go about solving this problem.
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Partial sum formula of a polynomial series?

I am trying to find the partial sum formula of the following series: $$ \sum_{y=1}^{\infty} \frac{4y^2-12y+9}{(y+3)(y+2)(y+1)y} $$ I have tried using Faulhaber's formula without success. I have also ...