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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Are there any good books on infinity?

I am looking for a book that discusses the concepts of infinity like the actual infinite and related concepts. Any suggestions?
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Doubts about infinite nested root

Find $f(a)=\sqrt{a-\sqrt{a^2-\sqrt{a^4-\cdots}}}$ where $a\in\mathbb{R}$. My Attempt : I consider $\frac{f(a)}{a}=\sqrt{1-\sqrt{1-\sqrt{1-\cdots}}}$. Now to finding this limit is easy but I cannot ...
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302 views

If $y=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$ then how $y=x^y$?

In questions like, find the derivative of $f(x)=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$, how can we formally show that $y=x^y$? We use this technique for all type of iterations, e.g. ...
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How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
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3answers
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Applying L'Hôpital's rule infinitely

I tried to prove that $\int\limits_0^\infty t^{x-1} e^{-t} \, \mathrm{d}t$ satisfies the functional equation of the gamma function $\Gamma(x+1)=x\Gamma(x)$, so I partially integrated $\Gamma(x+1)$, ...
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2answers
71 views

How to explain indeterminations, and some aprpoaches to $+\infty$ or $-\infty$, for middle school students?

Question: how to explain the undefinitions $0^0$ and $\frac{0}{0}$ for Middle school students?? I am a math teacher and I don't know how to answer properly when studens ask me why some operations ...
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1answer
161 views

Is there a highest order of infinity?

Does there exist an infinite set of cardinality such that it can never be reached by taking power sets of a set with cardinality aleph-null. Please prove your answer, or include a link to a proof. I ...
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1answer
30 views

Prove that function f has a local minima and maxima

$f:R->R, f(x) = (x^2+mx)e^-x$ Show that, for every m in R, the function f has a local ...
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1answer
91 views

What is the point of (Compactness Theorem in the) Overspill Principle?

I am trying to understand the basics of computation theory. The Overspill Principle (also at google) basically says if you are cool you can do everything Let Г be a sentence of predicate logic ...
2
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2answers
53 views

Expected value of a the reciprocal of a random number

If I selected a real number at random from the interval (0.0,1.0), assuming a uniform distribution, the "expected value" would be 0.5. (I am not certain I am using the phrase correctly; I mean, if I ...
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2answers
322 views

Is infinity a real or complex quantity?

Since I was interested in maths, I have a question. Is infinity a real or complex quantity? Or it isn't real or complex?
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Calculate $\displaystyle \lim_{x \to \infty} x - \sqrt{x^2 + 2x}$ without derivations.

How can I calculte $\displaystyle \lim_{x \to \infty} x - \sqrt{x^2 + 2x}$? Here is what I´ve done so far: Multiplying by $\displaystyle \frac{x + \sqrt{x^2 + 2x}}{x + \sqrt{x^2 + 2x}}$ I got ...
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0answers
93 views

Limit of constant function

I was reading a proof using Markov Chains in a finite state space $E$. Denote $p_{ij}(n) = P(X_n = j | X_0 = i)$. Since the state space is finite, then probability of landing somewhere in the state ...
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2answers
71 views

Difference between limits $\infty$ and $+\infty$

Is there a difference between these two limits? $$\lim_{x\rightarrow\infty}f(x)=+\infty\text{ and }\lim_{x\rightarrow+\infty}f(x)=\infty$$
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8answers
311 views

How come $1^{\infty}$ = undefined, while $2^{\infty} = \infty$ and $0^{\infty} = 0$? [duplicate]

$1^\infty$ = undefined $2^\infty = \infty$ $0^\infty = 0$ Why is $1^\infty$ undefined? People were trying to explain to me that infinity isnt part of the Real numbers, yet, $2^\infty$ and ...
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240 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
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3answers
164 views

What's wrong with using algebra on infinite series?

I've recently found an article (referred somewhere on this site) criticizing the use of common rules of algebra on infinite series. To be honest, the video referred is one of the videos of Numberphile ...
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1answer
277 views

Infinite Series of the asymptotic expansion of Fresnel Integrals

I need to find the infinite series for the asymptotic expansions of the fresnel integrals as $x\rightarrow \infty$ and $x\rightarrow 0$. Now I have computed the asyptotic expansions to be as follows ...
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5answers
74 views

How can infinity be an accumulation point?

I can't wrap my mind around this: An accumulation point is a point of a set, which in every yet so small neighborhood (of itself) contains infinitely many points of the set, right? So if (in my case) ...
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1answer
32 views

limits for positive and negative infinity

It says we use the l'hospital's rule, however I don't understand because the limit for positive infinity and negative infinity are different. Please help!
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1answer
36 views

How do we evaluate this limit?

$$ A_N(x) = \lim_{N\to \infty}(\sin(x)/x)^N $$ The solution to this problem is given as, $$ A_N(x) = \exp( -Nx^2/6). $$ The problem is solved through Taylor series expansion for $\sin(x)$. And ...
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2answers
79 views

A solution to the equation $\frac{1}{x}=0$ [duplicate]

The number $i$ is defined as a solution to the equation $x^2+1=0$. How come no one has yet defined a number $j$ as a solution to the equation $\frac{1}{x}=0$? The purpose of course is to be able to ...
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2answers
32 views

Find any sequence that meets these criteria.

I'm struggling with this problem and don't know where to start looking: Is there any sequence $a_n$ such that $\lim\limits_{n \to \infty}a_n \neq 0$ and $\lim\limits_{n \to \infty}(n \sqrt[n]{|a_n|}) ...
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Is it possible to extend the complex plane affinely and projectively at the same time?

Is it possible to extend the complex plane affinely and projectively at the same time? That is by adding both the positive infinity (with based on it directed infinity) AND the unsigned complex ...
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4answers
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What is the answer to the paradox of the infinitesimal?

I just read this article on npr, which mentioned the following question: You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're ...
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1answer
45 views

Does this relative primes formula violate the feasibility of picking truly random numbers?

I read a question on this site recently that fascinated me by pointing out that you can't truly pick a random number from an infinite set. I can't find the answer now, but it was shown that you have ...
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3answers
85 views

Can an infinite set be transitive, irreflexive, total, and have an upper and lower bound?

I need an infinite structure that can be put into an order with the following properties: The order must... be transitive, be irreflexive, be total (i.e., every two things share some sort of ...
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2answers
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Having trouble showing the cardinality of two infinite sets is the same

We just learned about Aleph-naught today and I read about it on wikipedia but I do not know how to go about solving this problem in my homework: Prove that N(natural numbers) has the same ...
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5answers
185 views

Why can't the reals be constructed from the infinitesimal?

If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then Why can't the reals, which demands, simply, unlimited precision (this ...
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2answers
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Cardinality of the set of at most countable subsets of the real line?

I'm exploring an unrelated question about power series with complex coefficients. While exploring this question, I wondered: What is the cardinality of the set of all such power series? Or with ...
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24 views

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 ...
0
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1answer
132 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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0answers
28 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
273 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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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 ...
0
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1answer
59 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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212 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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0answers
43 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
70 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
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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
91 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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2answers
158 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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76 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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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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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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237 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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485 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
95 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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136 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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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 ...