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?

learn more… | top users | synonyms

1
vote
1answer
106 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 ...
3
votes
1answer
54 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 ...
1
vote
3answers
159 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 ...
2
votes
3answers
219 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 $\aleph_1$.
1
vote
3answers
75 views

How to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$?

I need to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$.
3
votes
1answer
88 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 ...
3
votes
2answers
41 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, ...
2
votes
2answers
45 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 ...
5
votes
4answers
870 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 ...
-1
votes
1answer
92 views

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 ...
19
votes
4answers
697 views

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 ...
0
votes
1answer
43 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 ...
7
votes
4answers
146 views

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 ...
0
votes
2answers
143 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)? ...
0
votes
2answers
111 views

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} = ...
5
votes
2answers
143 views

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 & ...
2
votes
4answers
149 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 ...
0
votes
1answer
31 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 ...
-1
votes
2answers
308 views

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 ...
0
votes
1answer
84 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 ...
5
votes
3answers
246 views

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, ...
3
votes
1answer
81 views

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 ...
4
votes
0answers
73 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 ...
0
votes
2answers
46 views

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 ...
5
votes
5answers
454 views

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 ...
1
vote
2answers
31 views

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?
0
votes
0answers
53 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 $Y$ ...
3
votes
1answer
272 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 ...
1
vote
1answer
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, ...
2
votes
4answers
67 views

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?
2
votes
5answers
405 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.
1
vote
2answers
83 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 ...
0
votes
1answer
70 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?
0
votes
3answers
39 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. $\lim \limits_{x \to \infty} \sqrt{x^2+3x} - \sqrt{x^2-2x}$ ...
0
votes
1answer
41 views

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
1
vote
2answers
72 views

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. ...
0
votes
3answers
50 views

The limit of $\sqrt{x^2+x+1}-\sqrt{x^2+1}$ as $x\to\infty$ [closed]

Currently I'm self studying limits. but I don't know how to get the answer to this question: $$\lim _ { x\to \infty }\left(\sqrt{x^2+x+1}-\sqrt{x^2+1}\right)$$ can someone help me
1
vote
1answer
33 views

What is the cardinality of all frames in time?

If we divide time into individual frames, then we would get a set of infinite frames. But what is the cardinality of such a set? Since time is continuous, like the real numbers, I would expect the ...
1
vote
0answers
111 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. ...
0
votes
2answers
84 views

Limit that fail to exist

Does a limit that equals to infinity considered to exist ?? am confused !! for Example 1/(x-2)--> when evaluating the limit at 2 the result is 1/0 which is infinity while after looking at the graph ...
0
votes
1answer
83 views

Is$\ +\infty$ greater than any other number (surreal, superreal, hyperreal, …)?

Let$\ \mathbb{A}$ be an arbitrary totally ordered set and consider the largest element of the set of extended real numbers,$\ +\infty$. Can we say that$\ +\infty > \chi $, for *any*$\ \chi \in ...
1
vote
2answers
45 views

Question about $\lim_{x \to -\infty}\frac{\sqrt{10+11x^2}}{12+13x}$

$\lim_{x \to -\infty}\dfrac{\sqrt{10+11x^2}}{12+13x}$ = multiply top and bottom by $\dfrac{1}{x}=-\dfrac{1}{\sqrt{x^2}}$ My question is, why is the negative sign in front so crucial, I don't ...
0
votes
1answer
30 views

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$. My attempt: Let $x_n\to -\infty$ and $\epsilon\gt 0$. By the Archimedean Principle pick $N\in \mathbb N$ such that ...
1
vote
3answers
65 views

How do I solve $\lim_{x\to -\infty}(\sqrt{x^2 + x + 1} + x)$?

I'm having trouble finding this limit: $$\lim_{x\to -\infty}(\sqrt{x^2 + x + 1} + x)$$ I tried multiplying by the conjugate: $$\lim_{x\to -\infty}(\frac{\sqrt{x^2 + x + 1} + x}{1} \times ...
0
votes
3answers
69 views

Does this sequence diverge to ∞?

The sequence $(a_n)_{n \geq 1}$ is defined as follows: $$a_n:= \begin{cases} 0 \quad \text{if} \quad n \quad \text{is odd}\\ n \quad \text{if} \quad n \quad \text{is even}\end{cases} \quad .$$ Does ...
19
votes
7answers
3k views

There is no smallest infinity in calculus?

Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess. From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and ...
1
vote
3answers
380 views

Calculate exact value of and infinite sum [duplicate]

Im trying to find the exact value of the infinite sum : 3 + 1/3 + 1/27 + 1/243 + 1/2187 + ... I can see that to generate new terms we take the previous term and divide by 9 or multiply by 9. Not ...
2
votes
0answers
49 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 ...
0
votes
0answers
56 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 ...
3
votes
3answers
144 views

Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...