Questions tagged [infinity]
Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.
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Number of solutions of a trigonometric equation
In this question, all angles are expressed in radians.
The precise number of solutions for the unknown $x$ of the equation $$(\sin x+\cos x) \left( \sin x -\cos \left( \frac \pi 2 -x \right) \right) =...
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2
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How to prove that an infinite series does not converge to $0$?
I am now considering the following infinite series.
$$ \lim_{k \to \infty} \sum_{j=1}^{k} \frac{1}{j(j+1)} \left\{\exp\left(\frac{2 \pi i}{n} \right)\right\}^{j} $$
Here, $n \geq 2$ is a natural ...
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Cantor diagonalization argument [closed]
How do we exactly choose real numbers in (0,1) in decimal expression (a1,a2,a3..., b1,b2,b3... for 0.a1a2a3... and 0.bb1b2b3...) in each row of table, are those all real numbers in that interval? I've ...
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Limits involving points at infinity in the extended complex plane
Im trying to understand the idea of infinity in imaginary numbers.
In the book "Complex variables and applications" (8 edition, brown/churchill, section 17) to explain this concept the ...
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What are examples of models for which the Continuum Hypothesis is true/false?
I'm not a set theorist so pardon my improper language. I'm trying to make sense of the unprovability of the Continuum Hypothesis. What I've come to understand is this: since set theory is broad and ...
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Two questions about an infinite series [duplicate]
The infinite sum of the geometric series $$\sum_{k=0}^\infty\frac1{2^k}.$$ To think about this geometrically, let's call each $k$-th term a "segment" that gets added together to eventually ...
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GL2(F2) group acting on an infinite element
We are given a group $ GL_2(F_2)$ acting on the set $F_2 \cup \infty $ by fractional linear transformation $g•z=\frac{az+b}{cz+d}$. Here $F_2=\{0,1\} $. One of the elements of the group $ GL_2(F_2)$ ...
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2
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55
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Formal proof that an infinite complete binary tree has countably many infinite nodes
I want to prove that an infinite complete binary tree has countably many infinite nodes. However, most of the tried I made at this were developed from a computer algorithm, such as breadth first ...
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Trouble in understanding the result of $\lim\limits_{x \rightarrow 0} \frac{x}{0}$ as shown on Wolfram Alpha [closed]
So in wolfram alpha it says that $\lim\limits_{x \to 0} \frac{x}{0} = \infty^\sim$ where $\infty^\sim$ symbol is complex infinity.
I find it hard to understand this symbol and the concept, can someone ...
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Is the cardinality of an infinite set correlated with the amount of information needed to specify any one element?
I was thinking about the fact that some infinities are 'bigger' than others. For example, the cardinality of the reals is literally bigger than the cardinality of the rationals. One cannot match them ...
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Is there a Julia fractal that contains uncountable many copies of itself?
We know that the Mandelbrot fractal contains a countable number of copies of itself.
See :
Does the Mandelbrot fractal contain countably or uncountably many copies of itself?
Where that is explained.
...
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Extended Hilberts hotel
i just checked out a video about the "Hilbert's Hotel Paradox" and about it running out of room if you want more info checkout Veritasium's video(How An Infinite Hotel Ran Out Of Room the ...
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In an infinite sum, is there an actual term at an infinite position? [duplicate]
The sum of
$$
\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n = 1,
$$
exactly.
It has been proved that the sum does not just tend to 1 and that it is not just defined as 1, but rather, it is exactly ...
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Questions about the Infinite Monkey Theorem
(Context: the Infinite Monkey Theorem stipulates that given infinite time, a monkey can type out the complete works of Shakespeare, or any other text of finite length, just by randomly pressing keys.)
...
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Sum of reciprocals of $10$ to the power of factorials
$$\sum_{i=1}^{\infty} \frac{1}{10^{i!}}$$
I don't remember the name.
Also, if someone would be kind enough to send the proof of its transcendence.
Thanks, envy.
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$\lim\limits_{n\to\infty} \dfrac{0}{\left(\frac{2}{n^2}\right)^2}\overset{?}{=}0$. $n=\infty$ is not a case I need to consider, right?
Consider the sequence defined by $(x_n, y_n)=(\frac{1}{n}, \frac{1}{n}),\quad n\in \mathbb{N}$ and the function
$f(x,y)=-\dfrac{y^4+x^3y-xy^3-x^4}{\left(y^2+x^2\right)^2}$
Now I want to calculate the ...
2
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2
answers
182
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Uncountable Summation of Zeros [duplicate]
A peculiar line of philosophical inquiry drew me to this question: “Is the summation of an uncountable number of zeros, zero”? I am not very familiar (read as “basically know nothing”) with ...
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Comparing cardinality of set of real numbers $[2,3]$ and set of real numbers $[4,5]$
The function $f(x) = x+2$ on the given sets $([2,3]$ and $[4,5])$ is bijective which should imply that the cardinality is equal.
Now, consider the function $f(x) = x^2$ on the sets of real numbers $[2,...
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Math that does not have infinity
I am not a mathematician. So I am not even sure if what I am asking is logically coherent. But I do have some application-based curiosity that I would like to enlighten myself about. I will first pose,...
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Validity of the expression $\operatorname{arccot}(\cot(\pi k))=0$ by extending the definition to evaluate at discontinuities
I'm exploring the properties and behaviors of trigonometric functions, specifically the cotangent function $\cot(\pi x) $ and its behavior near its discontinuities. The normal domain of $\cot(x)$ ...
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2
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Zero to the power of negative numbers
I lately stumbled unto a strange behaviour of my calculator (a TI-Nspire™ CX II-T CAS):
$$
\begin{split}
0^{-2} = 0^{-4} &= \infty & \forall \text{ even numbers}\\
0^{-1}, 0^{-3} &=...
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1
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103
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Infinitely decreasing exponents
Consider the following expression: $$x=a^{{\frac{1}{2}}^{{\frac{1}{4}}^{\frac{1}{8}...}}}$$
Where $a$ can be pretty much anything. Normally, such exponents are evaluated from top to bottom. But this ...
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65
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Is $|\mathbb{R}^{\infty}| =|\mathbb{R}|$ [duplicate]
Is $|\mathbb{R}^{\infty}| =|\mathbb{R}|$
I started reading Rudin's principle of mathematical analysis and I read this theorem
$2.13$ Theorem Let $A$ be a countable set, and let $B_n$ be the set of ...
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The primes have cardinality $\aleph_0$. How does their powerset also have cardinality $\aleph_0$?
Given that the cardinality of the naturals is the smallest infinite cardinal ( $|\mathbb{N}| = \aleph_0$ ), and the primes $\mathbb{P}$ are an infinite subset of the naturals $\mathbb{N}$, we know ...
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Approximate estimation or evaluation to find the answer of the limit
As the title suggests, how can I find the answer of this limit?
I do not know exactly about what kind of this problem belongs to, maybe estimation or evaluation.
$$\lim\limits_{n \to \infty} n \left[(\...
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Evaluating $ \lim_{n \to \infty} (2^n + 6^n)^{1/n}$ [closed]
I'm completely stuck on how I would go about solving this problem. I'm not sure if I need to use e or ln. Could someone give hints on how I would go about solving this problem?
$$
\lim_{n \to \infty} (...
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1
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Question regarding validity of argument involving "infinitary construction" of a set
I apologize if the title to this post is not too clear. I was reading Jech's Set Theory and I came across Theorem 2.8: "If $W_1$ and $W_2$ are well-ordered sets, then exactly one of the following ...
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Proof verification: $P(\mathbb{Z}^+)$ is uncountable
I construct an injection from $(0,1) \setminus \mathbb{Q}$ to $P(\mathbb{Z}^+)$.
Here's basically how it works:
Take an irrational, say
$$0.214560000012511...$$
Split it into strings of digits like ...
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Minimum sum between permutations of elements in a matrix
So, for the purposes of creating a heuristic for a Sokoban AI, I have to solve the following problem (if you're curious, the elements of the matrix each represent taxicab distances from boxes to ...
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Infinite recursion of function defined with respect to "every $n$" [closed]
I have very little formal mathematical training, so I apologize in advance for what may seem like basic issues in this question's phrasing.
I am trying to determine whether I can make a certain ...
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What is the underlying meaning of LxWxH=Volume? [duplicate]
Say a rectangular solid has length=1, width=3, and height=5. Its base area would be 1x3=3. Its volume would be 5x3.
Now, 5x3 means 3+3+3+3+3, and each 3 represents the area of a 2D plane. So does this ...
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Does an infinite line segment make any sense? [closed]
In school, I was taught that a line segment must be finite. But a fractal has an infinite perimeter. If you just took two points along that perimeter, and "stretched it out" wouldn't you ...
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When can we "safely" use extended real numbers
I learned that we can define extended reals and some related arithmetic operations. I wonder when can we use these notations without worrying about consistency.
For example, if $\lim_{n\to\infty}a_n=\...
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3
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choosing a random integer from $\mathbb{N}$
I heard that if we choose a random positive integer, no matter how big it is, it will be closer to $0$ than $∞$.
I have an interesting question: if we assume that we had an infinite computer which had ...
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How would Cartesian Plane look at infinity?
I’m relatively new to mathematics, and I admit I’m still trying to grasp the concept of infinity, so there might be some errors in my thinking.
I understand that $\infty$ is not a real number, which ...
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Prove this wonderful trigonometric limit $\lim_{n \rightarrow \infty} (\cot(\frac{x}{n+1})-\cot(\frac{x}{n-1}))=\frac{2}{x}$
Show that $$\lim_{n \rightarrow \infty} (\cot(\frac{x}{n+1})-\cot(\frac{x}{n-1}))=\frac{2}{x}$$
One way to prove is using series but I am wondering is there any other way to prove it . Can anyone ...
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Evaluating the terms of basel problem upto given accuracy.
We know that $S = \sum _{n=1}^{n=\infty} \frac{1}{n^2} = \frac{\pi^2}{6} = 1.64493406682264364..$. By the definition of convergence, there exists some $N_0 \in \mathbb{N}$ such that $S_n > 1....
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show that $\lim_{x\rightarrow-\infty}\frac{1}{x^k}=0$
I am reading Calculus, Purcell and there is a solved example
Show that if $k$ a positive integer, then
$$\lim_{x\rightarrow\infty}\frac{1}{x^k}=0\ \text{ and} \ \lim_{x\rightarrow-\infty}\frac{1}{x^k}=...
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1
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Does this algebra have a name?
I thought of viewing $\bar{\mathbb{R}}$ as a relative subalgebra of a total algebra $\mathbb{R}^\ast := \bar{\mathbb{R}} \cup \{\ast\}$, wherein an output of an operation, if not already defined in $\...
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$x^y=y^x$: prove that y approaches $1$ as x approaches $\infty$ [closed]
$y^x=x^y$
I randomly graphed this interesting function. I have two question about it:
Its graph consists of two curves, one of which is $x=y$ and the other curve looks like a hyperbola(in the first ...
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Prove that $\lim _{x\to -\infty}x^n=\infty$ for $n\in \Bbb N$ and $n$ even, and $\lim _{x\to -\infty}x^n=-\infty$ for $n\in \Bbb N$ and $n$ odd
Prove that $\lim_{x\to -\infty}x^n=\infty$ for $n\in \Bbb N$ and $n$ even, and $\lim_{x\to -\infty}x^n=-\infty$ for $n\in \Bbb N$ and $n$ odd.
I tried solving this as follows:
If $n$ is odd, then we ...
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Find the value of $0.5^{0.5^{0.5^{0.5^\ldots}}}$
Find the value of $0.5^{0.5^{0.5^{0.5^\ldots}}}$
My attempt:
$$\begin{aligned}
&0.5^{0.5^{0.5^{0.5^\ldots}}} =: x \\
&\Rightarrow\ 0.5^x = x \\
&\Rightarrow\ -x \ln(2) = \ln(x) \\
&\...
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Limits Involving Infinity
In calculus, when we encounter limits of the form “A/0” where A is non-zero, this is not an indeterminate form and by observing the functions, most of the time each side blows up to $+ \infty$ or $-\...
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Is there a concept of finiteness independent of the successor function?
Why is there no infinite natural number, and why does finiteness need to be closed under the successor function?
I think can understand why something like $…S(S(0))…$ is not a natural number because ...
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1
answer
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Can we simplify the classification of infinite Dedekind-finite subsets of $\mathbb{N}$? [closed]
In ZF, does there exist a bijection between the infinite subsets X of $\mathbb{N}$ for which there exists no injection $\mathbb{N}\rightarrow X$ and the set of infinite Dedekind-finite subsets of $\...
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1
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96
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Are the "flipped" 10-adic number equivalent to the real segment $[0;1]$ [duplicate]
I just came with an question/idea trying to really feel why real numbers a uncountably infinte. I know Cantor's diagonal argument and I'm quite convinced by it, but here is the idea I had which has to ...
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Proof of cantor's theorem.
Claim: $P(\mathbb{Z}^+)$ is uncountable.
Proof.
Suppose its not; then we can let $X_1,X_2,...$ be an enumeration of $P(\mathbb{Z}^+)$. Let $g:P(\mathbb{Z}^+)\setminus\{\mathbb{Z}^+\}\to\mathbb{Z}^+$ ...
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1
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Hilbert's Grand Hotel is always hosting the same infinite set of guests
I am learning the fundamentals of mathematics.
A bit background: This article says that "The mathematical paradox about infinite sets" envisages Hilbert's Grand Hotel:
"...a hotel with ...
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230
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How can i evaluate the product? [closed]
I have to evaluate the following product:
$$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64} $$
But I couldn't develop anything. I thought about breaking the product into smaller parts with some ...
1
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1
answer
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Hilbert's fully occupied Grand Hotel
Q: Suppose that a countably infinite number of buses, each containing a countably infinite number of guests, arrive at Hilbert's fully occupied Grand Hotel. Show that all the arriving guests can be ...
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