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

learn more… | top users | synonyms

0
votes
3answers
66 views

On reading the notation $x\to \infty$

Why is it better to read $x\to \infty$ as "$x$ grows without bound" rather than "$x$ approaches/comes closer to infinity"?
5
votes
1answer
151 views

Are infinite products commutative?

While reading a textbook, I came across the following proof (for integer partitions into odd parts and distinct parts): The following steps can be justified by taking finite products and then ...
3
votes
3answers
207 views

cantor's diagonalization argument (multiple sizes of infinities)

Yes, many people have asked the same question, and I have tried by best to get my head around it, but I still can't understand. Every time I look at the proofs, they seem like a sleight of hand to ...
4
votes
2answers
96 views

Which texts do you recommend to learn about the work of Cantor?

A friend asked me if I know texts which talk about Cantor's Work on infinity and Set theory . And this is the reason Which pushed me to ask the question here . I prefer if you recommend texts for ...
2
votes
1answer
102 views

What is $\lim_{x=0\rightarrow1}x^\infty$?

I know it's all about definition... But still I want to know whether the answer is $0$, $1$, impossible to say or something else, like that the mathematical statement is wrong. However, to clarify, ...
1
vote
2answers
255 views

How can a function's range be 'the reals including infinity'

In this video the man says that `a measure $\mu$ on $\Omega$ with $\sigma$-algebra $\mathscr{A}$ is a function $\mu \colon\mathscr{A} \rightarrow[0,\infty]$ s.t. [...]. What does that mean? The part ...
10
votes
5answers
11k views

Prove that the distance between a black and a white dot is one

I just read this article about some tough interview questions. One of the questions (allegedly given in an interview for a Technology Analyst position in Goldman Sachs) was: There are infinite ...
1
vote
1answer
159 views

$\pi$ and $\ln4$ relations. Even and Odd alternating sums.

Tonight, playing around on WolframAlpha, I discovered that the alternating sum of the odd numbers is $\frac\pi4$ and the alternating sum of the even numbers is $\frac{\ln4}4$ Are there any known ...
1
vote
2answers
683 views

Comparing infinite binary fractions to infinite decimal fractions

I'm trying to understand the cardinality between the set of all infinite binary (base-2) fractions and the set of all infinite decimal (base-10) fractions. I can easily think of infinite binary ...
7
votes
4answers
2k views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
3
votes
7answers
675 views

When does it make sense to say that something is almost infinite?

I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference. As someone who hasn't ...
0
votes
2answers
79 views

Limit approaching infinity-related question

Why is $$\lim_{x\to\infty}\frac{x^2}{1+x^2}=1?$$
1
vote
1answer
579 views

zero raised to infinity

I encountered a question where the only condition stated that $t>0$ and was then asked to compare these two quantities $0^t$ $t^0$ The scope of $t$ is $(0,\infty)$ and hence for infinity 1.) ...
0
votes
3answers
560 views

Equinumerosity of infinite sets

Key issue: For infinite sets, does the existence of a bijection mean they have the same number of elements? For example, does the set of natural numbers N = {1,2,3,4...} have the same number of ...
1
vote
4answers
148 views

Infinity and structures

Do you know any case (example) where an "infinite" object with a structure (say, an infinite group) cannot be extended (in the sense of adding elements) in any way without it no longer having the ...
3
votes
2answers
600 views

Proof of a Property of Vertical Asymptotes

I'm trying to understand a proof in my Calculus textbook of the following theorem: Let the functions $f$ and $g$ be continuous on an interval containing $c$. If $f(c) \neq 0$, $g(c) = 0$, and ...
1
vote
1answer
353 views

Are the number of terms in an infinite series even or odd?

This question arose after I saw a youtube-vid where Grandi's series was discussed.It seems that the sum of the series will be 0 for an even, and 1 for an odd number of terms, where a term is defined ...
4
votes
1answer
85 views

Generlized Büchi Games and Closed under superset Muller Games

For a unique infinite play $p$ in a 2-Player game $G=(V_0,V_1,E)$. Let $$ \inf(p) \subseteq V_0 \cup V_1 $$ be the set of vertices which occur infinitly often in $p$. Generlized Büchi (GB) Games ...
5
votes
1answer
261 views

How many centers in a infinite connected graph?

I find a very interesting concept "center" while learning basic graph theory. A center of graph $G$ is a vertex with the minimal greatest distance (eccentricity) to other nodes in $G$. Now I’m curious ...
2
votes
1answer
77 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 ...
2
votes
1answer
122 views

Infinite Parity Function

I was looking at this problem, and I have a solution for a finite board with $2^n$ squares, that I want to extend to a countably infinite board. Label the squares from $0$ to $2^n-1$. Consider the ...
2
votes
2answers
244 views

Can you prove that function is positive for certain values using limit of function?

Here's example: Prove that $x-\frac{1}{x^2} \geq 0$ for every $x \ge 1$ I know that this can be done using elementary algebra, but in other cases it's not that simple. Can I prove this inequation ...
0
votes
2answers
637 views

Maximum Number in an Infinite set

Given an infinite set of random integers, is there a largest element? In other words is maximum as a concept inherently tied to finite sets?
9
votes
2answers
5k views

What's the difference between Complex infinity and undefined?

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!
14
votes
2answers
328 views

Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
-2
votes
6answers
180 views

Why is $1 =1 $? why is it so why cant be $1 =$ something else? [closed]

It may sound stupid but why is $1=1$ or $n=n$ if thats the case does $1/0 = 1/0.$
3
votes
2answers
202 views

Are real numbers also hyperreal? Are there hyperreal $\epsilon$ between $-a$ and $a$ for any positive real $a$?

The set of all hyper-real numbers is denoted by $R^*$. Every real number is a member of $R^*$, but $R^*$ has other elements too. The infinitesimals in $R^*$ are of three kinds: positive, negative ...
1
vote
0answers
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 ...
10
votes
7answers
424 views

Find $\lim_{x\to-\infty}{x+e^{-x}}$

I have this exercise in my worksheet: $$\lim_{x\to-\infty}{x+e^{-x}}$$ I am always ending up with $-∞+∞$ or $\frac{∞}{∞}$. It says the answer is $+∞$, but how can I get that?
4
votes
2answers
427 views

how to know the set is finite, countable or uncountable

I am trying to understand whether the set is finite, countable, or uncountable. $$\{x \in\Bbb Q \mid 1<x<2 \} \qquad\text{is countable. }$$ but i dont understand why though. is it countable ...
3
votes
1answer
351 views

Why does Michio Kaku say that $\frac{1}{0} = \infty$?

Why does Michio Kaku say that $\frac{1}{0} = \infty$? http://youtu.be/AJ4zlvqOtE8?t=4m43s Instead of $\frac{1}{0}$ that's not defined, so we don't know.
0
votes
1answer
372 views

Absolute value of infinite sum smaller than infinite sum of absolute values

A question emerging from an exercise in Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press. The exercise consists in showing that if $\sum_{i=1}^\infty x_i$ ...
3
votes
2answers
570 views

Taking the limit of an integral using residues, why is this wrong?

I have the integral $\lim\limits_{R\to\infty}\int_{-R}^{R} \frac{\cos(x)}{x^2+a^2} dx$ where $a$ is a positive real number. The strategy was to evaluate the limit of the integral on the boundary of a ...
4
votes
2answers
426 views

Singularities of $f(z)=z/\cos(z)$

Regarding complex functions (in complex variables), I was wondering why the function $g(z)= \cos(z)$ has a singularity at $z = \infty$ but $f(z)= \dfrac{z}{\cos(z)}$ does not. I am a bit confused ...
7
votes
1answer
199 views

What happens to the infinite monkey theorem when there are an infinite number of keys on the typewriter?

What happens to the infinite monkey theorem when there are an infinite number of keys on the typewriter? So what is the probability of a finite string of keys like the works of Shakespeare being typed ...
2
votes
2answers
244 views

asymptotic infinity

I am very very new to math as a whole, so please excuse my n00biness. I read: ...
4
votes
4answers
160 views

Dividing Two Infinities

I am Curious if the following is mathematically correct: Let $a$ be the infinite set of all nonnegative integers $0,1,2,3...$. Let $b$ be the infinite set of all nonnegative EVEN integers ...
2
votes
1answer
90 views

Is there an element in $^* \Bbb N$ is Dedekind-infinite?

One definition of a finite set is that it can be injected into an initial segment of $ \Bbb N$, thus any $n$ in $\Bbb N$ is finite. Accordingly, if it's legitmate to define every element in $^* \Bbb ...
-1
votes
3answers
224 views

Can countability coexist with infinity?

This question concerns the countability of the real numbers. First I will show how I count the numbers between 0 and 1 on the real line. It is done by reversing digits behind the coma, so that e.g. ...
10
votes
6answers
482 views

A question with infinity

I am a sophomore in high school and my math teacher did a very short lesson on infinity, here's how it went: (Try to solve each part yourself the first two are easy) Part 1 You have an inf. number ...
5
votes
2answers
601 views

Subtracting two infinities

I am Curious if the following is mathematically correct: Let $a$ be the infinite set of all nonnegative integers $0,1,2,3...$. I take from $a$ some of its elements, say integers $10$, $11$, and $12$ ...
2
votes
2answers
156 views

What is wrong with this proof that $\pi=\infty$

Does this "proof" show that $\pi =\infty$??? http://www.academia.edu/1611664/Sum_of_an_Infinite_sequence_PAPER Is there something wrong with the subbing in of infinity at the end of the paper?
4
votes
2answers
116 views

Infinite shots fired in a lattice forest

A hunter is standing in the center of an infinite 2D forest. There are point trees at all the integer lattice points. The hunter fires a gun with a bullet of zero width in a random direction. He ...
1
vote
3answers
320 views

How big is the size of all infinities?

"Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity..." What does exactly mean? How many infinities are there? I've heard there are ...
2
votes
1answer
79 views

What are the factors of $\aleph_0$?

Extend the system of positive natural numbers with $\aleph_0$. Then we have: $$\aleph_0 = \aleph_0\cdot n,\quad \forall n \in \mathbb{N}^+$$ Does it make sense to talk about factors of $\aleph_0$? ...
31
votes
5answers
2k views

Which infinity is meant in limits?

For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why?
2
votes
4answers
1k views

infinity times infinitesimal - what happens?

So what happens if we multiply infinite number by. Infinitesimal number? Like $dx \times \infty$ where $dx$ is treated as in one-dimensional integration. Also, can we divide infinite number by ...
6
votes
3answers
247 views

Use of infinity as an “idealistic approximation”

There have been several recent posts about the work of N. J. Wildberger, a finitist who seems to think that mathematics should only focus on things that have some sort of "real world" connection, ...
2
votes
6answers
348 views

prove that , $2+2+2+2+2+ \cdots= 1+1+1+1+1+\cdots$

how can we prove that ?? i think they are equal but a friend say that they are not equal my argument is $$1+1+1+1+1 + \cdots = \infty$$ $$2+2+2+2+2+\cdots = (1+1) + (1+1) + \cdots = (1+1+1+1+1 ...
4
votes
4answers
2k views

Non-existence of irrational numbers?

I realize the title of my question will probably cause the raising of some eyebrows, so let me explain. Not sure whether to file this under "math" or "philosophy". This also might be able to be ...