Tagged Questions

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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3
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2answers
75 views

Why are these expressions indeterminate expressions?

Why are these $1^\infty,$ $0\cdot\infty$ and $\infty^0$ indeterminate forms. Why we can't solve these expressions?
6
votes
3answers
539 views

What is the cardinality of the set of infinite cardinalities?

I am currently aware of only two infinite cardinalities: $\aleph_0 = |\Bbb N|$ $\aleph_1 = |\Bbb R|$ Questions: Is there an infinite number of infinite cardinalities? If yes, is this set of ...
0
votes
2answers
126 views

Infinite chessboard question

Suppose there is a species of aliens called cyborgs. There is an infinite chessboard in their homeland. There is 1 cyborg on every square. If cyborgs can jump infinitely far, or jump and land on their ...
0
votes
3answers
200 views

Infinity and Hilbert's hotel paradox

I did some infinite series calculations while studying Fourier analysis and the concept of infinity really bugs me. I haven't read or heard not one sensible explanation yet (for me), what infinity ...
8
votes
4answers
456 views

Why does $ (\frac{1}{2})^∞ = 0?$

Recently while at my tutoring I had a question that said: "Aladin has a pair of magic scissors that can cut things in to tiny pieces. If he cuts a carpet in half, cuts the half into half and continues ...
3
votes
2answers
92 views

Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$?

As the title says, my question is: Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$? I'm fairly certain this is true for finite sets but maybe ...
0
votes
5answers
132 views

Do non-square infinite matrices exist?

Sorry, I tried to wrap my head more around this, but I failed. Given non-square matrix $A$ that has dimension $kn \times n$. Now let $n$ goto infinity. Is the matrix finally square?
8
votes
2answers
94 views

Is $\frac{0}{0}$ different from $\frac{1}{0}$?

In my mind, zero divided by zero answers the question of what $a$, when multiplied with zero, equals zero: $a * 0 = 0$ Obviously, any real number will satisfy this equation. However, one divided by ...
0
votes
1answer
38 views

What is the cardinality of the equivalence class

Consider this relation: $$R = \left\{ {\left\langle {f,g} \right\rangle \in {{\left\{ {0,1} \right\}}^N} \times {{\left\{ {0,1} \right\}}^N}|\exists k \in N\left| {\left\{ {i \in N|f(i) \ne g(i)} ...
0
votes
1answer
125 views

$0.999999\ldots=1$? Others? [duplicate]

I have seen this problem come up many times and I was wondering if my proof is valid for $0.999\ldots=1$ where $0.999\ldots$ is continuous: $$x=0.999\ldots$$ $$10x=9.999\ldots$$ ...
1
vote
0answers
32 views

Naive calculations with infinite series [duplicate]

In the realm, where the sum of natural numbers is $-1/12$ : $1+2+3+4+...=-1/12$ Is this true?: $2+4+6+8+...=2*(1+2+3+4+...)=-2/12$ Can this kind of naive calculations always be done? -or are there ...
1
vote
2answers
42 views

Evaluating an integral over inifinty with polars leads to an integral of cosine over inifinity, how can this be resolved?

So I have the integral $$\int_0^\infty\int_0^\infty\frac{yx^2}{x^2 +y^2}e^{-(x^2 +y^2)} \,dx\,dy$$ And converting this into polars gives: $$\int_0^\infty r^2 e^{-r^2}\,dr ...
0
votes
2answers
61 views

Finding a limit with a Square Root

$$\lim_{x\to \infty} \frac{\sqrt{9x^6-x}}{x^3+7}$$ I thought it would simply be $1/3$, not sure where I went wrong.
0
votes
2answers
67 views

Are all infinite sets equivalent by an indexing function?

Would it be true to say that two infinite sets would always be equivalent since you could always match the index of their elements? For instance, match the first element in $A$ to the first element ...
0
votes
2answers
95 views

What is infinity to the power zero

Edited I have this notation $\lim_{k->\infty} k^ {1/k}$. Is it correct to say that the output is 1, or there is some other result. P.S: Okay guys made a mistake, sorry.Now please cool down. I am ...
0
votes
2answers
45 views

Show that any interval [A, B] of the number axis is equivalent to any other interval [C, D].

I am attempting to get my head around intervals, particularly the title question as described in What is Mathematics? (Courant & Stewart). I think I am probably misunderstanding the meaning of ...
1
vote
1answer
44 views

Cardinality of Orderings of $\mathbb{R}$

For a finite set $S$ there are $\vert S\vert!$ orderings of its elements. What is the cardinality of all orderings of $\mathbb{N}$? What would $$\vert \mathbb{N}\vert!$$ mean? Is it ...
5
votes
5answers
2k views

Does an equation containing infinity not equal 0 or infinity exist?

Does an equation containing infinity which is not equal to 0 or infinity exist? My math education stopped at poorly understanding trig so don't kill me please. OK so the question I meant to ask was ...
-1
votes
2answers
132 views

What digits is the “number” infinity composed of?

I have seen from past posts on the topic of infinity that there is some ambiguity with the concept infinity and whether it is a number etc. From what I can gather the terms number and infinity are ...
1
vote
1answer
34 views

Limits to infinity (n)

Hi I have a question regarding finding the values of limit for the following equation. The question states to find the following limits: $$ \lim_{x\to\infty}\left(\frac ...
3
votes
6answers
318 views

What is larger: The inside or the outside of the infinite circle? [closed]

Assume a circle with radius $R$ in a plane. Let $R$ go to infinity. What is larger: The inside or the outside of the circle? EDIT My naive way of thinking about "largeness" was just to compare ...
15
votes
3answers
2k views

Is an infinite line the same thing as an infinite circle?

Imagine that you are sitting next to a line that extends infinitely in both directions. Is it possible to distinguish it from an infinite circle? From my poor understanding of topology, I would ...
1
vote
5answers
101 views

Limits to infinity Finding Constant Number

Hi I have a question regarding of limits to infinity please help which I need to find the constant number for a and b. Please help! Thank You! The question states the user to find the following ...
1
vote
2answers
43 views

Find the Limit (n to infinity)

Hi I have a question regarding of limits to infinity please help! Thank You! The question states the user to find the following limit: $ \lim_{n\to\infty} n^2 ({\sqrt[n]{x}-\sqrt[n+1]{x}}) $ ...
2
votes
0answers
32 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} = ...
2
votes
1answer
60 views

Is this improper integral answer correct?

So I'm working on improper integrals and con/divergence and want some assurance that I've done the following correctly. $\int^∞_{-∞}cos(\pi t)$ As far as I'm aware this is convergent if and only if ...
4
votes
6answers
252 views

justification of a limit

I encountered something interesting when trying to differentiate $F(x) = c$. Consider: $\lim_{x→0}\frac0x$. I understand that for any $x$, no matter how incredibly small, we will have $0$ as the ...
6
votes
4answers
402 views

What happens if I toss a coin with decreasing probability to get a head?

Yesterday night, while I was trying to sleep, I found myself stuck with a simple statistics problem. Let's imagine we have a "magical coin", which is completely identical to a normal coin but for a ...
0
votes
1answer
58 views

Depth of infinite direct sum

Let $R$ is a local ring, from the depth lemma, we can get $\operatorname{depth}(R\oplus\dotsb\oplus R)=\operatorname{depth}(R)$, here the direct sum is finite, how about the infinite case? By the ...
1
vote
1answer
71 views

In an infinity of choices, is it possible to guess the correct one?

So I've been thinking about the infinite universes model, where each possible action or event creates a new universe for each outcome. For example, if you flip a coin there will be one universe in ...
0
votes
3answers
96 views

Epsilon-Delta Proof at infinity

Let $n \in \mathbb{N}$ and $y \in \mathbb{R}$ and $0<y<1$. Let also be $f(y)=y^n$ and $g(y)=y^{n+1}$. $$ \lim_{n \to \infty} \cfrac{f(y)}{g(y)} = L $$ What is the value of $L$ using the ...
0
votes
3answers
186 views

Is $\infty / \infty = 1$?

Lately, my friend and I were arguing about what $\infty / \infty$ equals. My thinking was that $\infty / \infty = 1$, since no matter how high you go in the numerator, it would have to go equally as ...
1
vote
2answers
105 views

Indeterminate form limits question

$$\lim_{x\to 0}\frac{10^x - 2^x - 5 ^ x + 1 } {x\tan x} $$ This is an indeterminate limit. I want help in solving this problem. Thanks in advance
0
votes
1answer
75 views

Countable and Uncountable sets

Is $\mathbb{N}\cup\{a\}$, for some $a\not\in\mathbb{N}$ countable or uncountable? $\mathbf{Attempt: }$ It is true that a set is countable if there exists an injective function $f : S → N$ from $S$ to ...
0
votes
0answers
36 views

infinite dimensional Cramer-Wold theorem

The Cramer-Wold theorem states that if every fixed linear combination of $d$ random variables converges to a normal distribution, then the $d$ variables jointly converges to a multivariate normal ...
1
vote
1answer
81 views

A Monkey Choosing Real Numbers for an Infinite Time

A common illustration of the nature of infinity is that, given an infinite amount of time, a monkey on a typewriter will, with probability $1$, produce the complete works of Shakespeare. Consider now ...
0
votes
1answer
171 views

Finding limit of a function as it approaches infinity

How do i solve the below without using L'hopital rule. The final answer obtained is $2/3$ ...
1
vote
2answers
107 views

Limit when an expoent goes to infinity

Please could someone help me and see if my solutions are correct for these two limits Let $n \in \mathbb{N}$ and $y \in \mathbb{R}$ and $y>0$. Case 1 $$\lim_{y \to \infty} ...
0
votes
1answer
37 views

Number of k-permutations that have odd number of an element

I want to find a recurrence relation $h_k$ for the number of k-permutations of $\{\infty a,\infty b, \infty c, \infty d \}$ that have an odd number of a's. I let $h_0=0$ because there is no odd ...
33
votes
6answers
2k views

Why is there antagonism towards extended real numbers?

In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept ...
2
votes
3answers
422 views

Why is the integral of sec^2(x) from 0 to pi infinity?

Why is it, if you take the integral of sec^2(x) from 0 to pi, my calculator returns "infinity" as the answer, but according to the second fundamental theorem of calculus, I got 0 with my own work. I ...
0
votes
2answers
105 views

Is this true that if limit approaches infinity the function equals to zero?

I would like to know if the following is true. If $$\lim_{z\to \infty} 1/f(z) = \infty$$ is that equivalent to $$\lim_{z\to \infty} f(z) = 0?$$
0
votes
2answers
46 views

Questions about hyperbolas and integration

I have a couple of questions regarding hyperbolas and their integrals. If it's too much, don't feel like you have to answer all 3 questions. My first question: The integral of a function like 1/x^2 ...
5
votes
10answers
2k views

Smallest next real number after an integer

This might be a silly question, but is it possible at all for n.00000...[infinite zeros]...1 to be the next real number after n? If not, why not? Firstly, I know (I think) that $$\lim_{x\to \infty} ...
-1
votes
4answers
146 views

Does $\infty^0=1$?

I was wondering if $\infty^0=1$. Some people have told me that there is no answer; it is undefined. Others have told me that the answer is $1$, using the rule $a^0=1, \ a\neq 0$. If it is truly ...
0
votes
0answers
28 views

Infinite One-Time Pad

As you know, when used correctly, a one-time pad allows one to send a message, such that the only thing that can be found out about it is the maximum size (which is also the key length.) It is ...
0
votes
2answers
38 views

Evaluating a limit as $x\to -\infty$

I am trying to evaluate $$ \lim_{x \to -\infty} \left(1+ \frac{1}{x}\right)^{x²}. $$ I'd say it tends to 0, 1 or something linked to $e$ but I have no clue how to prove this... I'm getting really ...
0
votes
3answers
37 views

Computing the limit of this function

So I have an improper integral: $$ \int_0^\infty \frac{13x}{x^2+1}-\frac{65}{5x+1} dx $$ I have solved the integral into this: $$ \lim_{t \to \infty} ...
14
votes
5answers
2k views

Is half a pie as big as a whole pie?

I am reading an e-book called To Infinity and Beyond by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole". In ...
0
votes
1answer
3k views

What is zero times infinity? [duplicate]

If any number times zero is zero and any number time infinity is infinity, then what do you get when you multiply zero times infinity? Do they cancel one another out and equal any number since any ...