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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How to properly clamp Beckmann Distribution

I am trying to implement the Cook-Torrance Microfacet BRDF shading model and I am having some trouble with the Beckmann Distribution: Beckmann Distribution with width parameter ...
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1answer
53 views

Are all uncountable infinities greater than all countable infinities? Are some uncountable infinities greater than other uncountable infinities? [duplicate]

I recently finished a discrete mathematics class, and near the end of the semester, the prof (very superficially) touched on countable and uncountable infinities. His explanation of countable ...
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6 views

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity $\infty$?

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity? Are there any paradoxes arizing? What are the rules when the ...
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3answers
77 views

Is infinity a real number? [duplicate]

Is infinity a real number? If not, why not? I want some very good arguments. Thanks. $$\rightarrow\leftarrow\Huge\Huge\Huge\boldsymbol\infty$$
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2answers
22 views

Equal number of 1s and 0s in number of n digits

How many ways could one create a binary number of n digits where the number of 1s and 0s are equal? For example, if n was 8 then we could have: 10101010 or 11110000 In addition to this, I may ...
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1answer
50 views

Calculus 2 Integral Question

I've been trying to resolve a calculus question and seem to be having troubles understanding exactly how to approach it. Some hints are supplied, but they don't exactly seem to help. Thanks to anyone ...
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3answers
107 views

Does the Mandelbrot fractal contain countably or uncountably many copies of itself?

I've been working on a program that draws fractal images, and I was struck by a question that came to mind. It is clear that the Mandelbrot fractal contains infinitely many copies of itself, but I've ...
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40 views

Help with calculating infinite sum

I'm working on a problem, and I'm stuck in the calculations of finding $\sum_{0}^{\infty}\frac{1}{1+n^2}$ Suggestions on how to approach this calculation? Thanks! (Also, I used Fourier to get to ...
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2answers
54 views

Difficult limit problem

$\lim_{n\to \infty} {\sqrt[n]{n^{-n}+2^n}}$ Intuitively, this seems like it should equal 2, but how would one go about showing this? I have tried factoring this somehow, but whatever form I get it in ...
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1answer
35 views

Is there an expression for $(S/k)$ where $S=\sum_{n=1}^\infty n$ and $k \in \mathbb{Z}$?

Given that $S=\sum_{n=1}^\infty n=-1/12$ (for an explanation see this question or this video from Youtube) For example if $k=4$: $(S/4)=1/4+2/4+3/4+1+5/4+6/4+7/4+2+9/4...$ Please edit to improve ...
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2answers
83 views

Is $\lim\limits_{n \to \infty} n$ “equal” to $\mathbb{N}$?

In set theory, the natural numbers are defined by means of inductive sets and the successor operation $S(n+1) = n \cup \{n\}$ As such, we have $1 = \{0\}$, $2 = \{0, 1\}$, $3 = \{0, 1, 2\}$, ...
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114 views

Is Continuum Hypothesis false? [closed]

The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers and the real numbers. However, it seems that it is possible to construct sets that ...
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2answers
67 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?
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3answers
451 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 ...
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2answers
110 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 ...
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3answers
72 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 ...
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4answers
403 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 ...
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2answers
85 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 ...
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5answers
120 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?
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2answers
77 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 ...
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1answer
35 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)} ...
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1answer
111 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: $$x=0.999\ldots$$ $$10x=9.999\ldots$$ ...
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0answers
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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 ...
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2answers
38 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 ...
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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.
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2answers
26 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 ...
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2answers
79 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 ...
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2answers
39 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 ...
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1answer
38 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 ...
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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 ...
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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 ...
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1answer
29 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 ...
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6answers
293 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 ...
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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 ...
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5answers
48 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 ...
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2answers
33 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}}) $ ...
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1answer
25 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} = ...
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1answer
43 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 ...
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6answers
234 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 ...
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4answers
342 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 ...
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1answer
46 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 ...
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1answer
53 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 ...
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3answers
75 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 ...
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155 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 ...
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2answers
62 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
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1answer
193 views

Series of numbers $1^2 + 2^2 + \dots$ [closed]

If we have a series of numbers $1^2 + 2^2 + 3^2 + .. +(10^n)^2$ ~ 1/3 * ${({10}^n)^3}$ and if we have another series of numbers $1^{29} +2^{29} + 3^{29} + ..+ (10^n)^{29}$ ~ 1/30 * $(10^n)^{30}$. ...
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1answer
56 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 ...
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0answers
15 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 ...
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1answer
63 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 ...
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106 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 $2/3$ ...