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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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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87 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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47 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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31 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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54 views

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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50 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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46 views

Is a nonzero number infinitely greater than zero? [closed]

So many years ago, I posted this question on Yahoo! Answers, and was not really happy with the response. I ran across it again recently and decided to try and breathe new life into this in the hopes ...
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45 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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58 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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36 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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3answers
73 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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Which infinity do we mean by $\infty$ in the symbol $x\rightarrow\infty$? [closed]

In ordinary mathematics we use the "limits" frequently. In principle the notion of "limit" is closely related to the notion of "infinity". Intuitively when we are calculating a "limit" we begin from ...
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116 views

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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63 views

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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1answer
71 views

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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38 views

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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41 views

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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756 views

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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35 views

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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Iterated limits (zero times infinity)

$\lim_{x \to 0} (x \cdot \lim_{y \to +\infty} y)$ Are there any rules considering these kinds of limits? I know the order of limits can't be interchanged in general. Can the infinity from the inner ...
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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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69 views

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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60 views

An infinite series question. [closed]

Ok, so we have an infinite sequence: S1 = 6+14+22+30+38.. Now, there is another infinite sequence, S2 = 1+2+3+4+5+6+7.. We know that the first sequence's nth term = (8n-2 ) so surely there must be a ...
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4answers
140 views

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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28 views

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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integration featuring the unit step function

Compute the following integrals I don't know how to use MathJaX so here's a link to the image of the integrals where u(t) is the unit step function and σ is some variable of integration
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37 views

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?

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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1answer
150 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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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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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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41 views

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 ...
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57 views

Is$\ \infty \times 0$ undefined in the extended real numbers?

And if it is, why? Is it a kind of postulate related to the fact that infinitely many points make a line?
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38 views

How can I evaluate this limit?

I'm studying for an upcoming midterm and i'm stuck on this question. It's asking me to evaluate the following limit and justify my answer. $\lim \limits_{x \to \infty} \sqrt{x^2+3x} - \sqrt{x^2-2x}$ ...
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What is the geometric way of relating zero to infinity?

I once saw (what I think was) a geometric way a relating zero to infinity. Something about a circle with radius 1 around the origin. Can you tell me where to find that? Thanks
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When is an infinite set larger than another infinite set?

Somewhat of a basic question that I've been pondering about, suppose we have 2 finite sets $A,B$, arbitrary sets with arbitrary elements that we know nothing about, except that they are both finite. ...