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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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1answer
19 views

Analytical convergence of an equation?

I am reading a paper which shows the following equation: $$ x_t = x_i + \alpha x_{t-1} $$ Where $x_i$ is the initial value, and $\alpha$ is a simple real constant. I am trying to ascertain if ...
1
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1answer
34 views

Comparing the number of real numbers between 2 ranges

I know that there are infinitely many number of real numbers between (1,2), (2,3) and so on and I know that there's no meaning to compare the number of real numbers between 2 ranges of real numbers ...
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3answers
68 views

Integrals of $\int^\infty _2 \frac{x}{x^3 -2\sin x}dx$

How do I calculate the convergence of $\int^\infty_2\frac{x}{x^3 -2\sin x}dx$ ? I know in generally, integration of (n-dimentional polynomial)/(n+2-dimentional polynomial) will be converge.
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0answers
22 views

Let $A$ be infinite and $\operatorname{Sq}(A)$ be the set of finite sequences in $A$. Then $A$ and $\operatorname{Sq}(A)$ are equinumerous

Let $A$ be infinite and $\operatorname{Sq}(A)$ be the set of finite sequences in $A$. Then $A$ and $\operatorname{Sq}(A)$ are equinumerous. My attempt: Let $A_n$ be the set of finite sequences of ...
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1answer
35 views

Let $(A_i\mid i\in I)$ be a family of disjoint sets where $|A_i|=|A_j|\ge\aleph_0\space\forall i\in I$. Is $|\bigcup\limits_{i\in I}A_i|=|A_i|$?

Let $(A_i\mid i\in I)$ be a family of sets where $|A_i|=|A_j|\ge\aleph_0$ and $A_i\cap A_j=\emptyset$ for all $i\neq j$. I found that we can prove $|\bigcup\limits_{i\in I}A_i|=|A_i|$ for all $i\in I$...
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2answers
77 views

Partition an infinite set into countable sets

Let $X$ be an infinite set. Find a partition of $X$ where each element in the partition is countable. Let $Y$ be the set of all families $\{F_i\mid i\in I\}$ in which each family satisfies 3 below ...
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0answers
66 views

If $X$ is infinite, then $X$ and $X\times X$ are equinumerous

If $X$ is infinite, then $X$ and $X\times X$ are equinumerous. Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help! My attempt: We denote $A$ and ...
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2answers
96 views

Why Hilbert changes the property of a set in his Infinite hotel?

I'm not a mathematician, so my question may look a bit lame to most of you. In the Infinite Hotel paradox we are dealing with infinite set of pairs (room/guest). A main property of a pair is 50/50 ...
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0answers
71 views

Let $X$ be infinite. Then $X$ and $X\times\Bbb N$ are equinumerous

Let $X$ be infinite. Then $X$ and $X\times\Bbb N$ are equinumerous. Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help! My attempt: We denote $A$...
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0answers
38 views

Suppose $X$ is infinite and $A$ is a finite subset of $X$. Then $X$ and $X\setminus A$ are equinumerous

Although the title is the same as one in Suppose $X$ is infinite and $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous, but the proof is different. Hence it's not a ...
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7answers
123 views

The limit of $\frac{n^3-3}{2n^2+n-1}$

I have to find the limit of the sequence above. Firstly, I tried to multiply out $n^3$, as it has the largest exponent. $$\lim_{n\to\infty}\frac{n^3-3}{2n^2+n-1} = \lim_{n\to\infty}\frac{n^3(1-\...
1
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1answer
32 views

An infinite division and balls problem

Suppose you have a bottle with infinite volume and infinite number of balls. Now it's 11 o'clock and an hour left till 12. So you put 10 balls in and take one out 30 minutes later. You repeat this ...
2
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0answers
40 views

Let $X$ be uncountable and $A$ be a countable subset of $X$. Then $X$ and $X \setminus A$ are equinumerous

Let $X$ be uncountable and $A$ be a countable subset of $X$. Then $X$ and $X \setminus A$ are equinumerous. I have previously proved that Suppose that $X$ is infinite and that $A$ is a finite subset ...
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4answers
732 views

Precise meaning of infinitely many

I have a rather lame question here. I need a clarification with the definition of "infinitely many". I have come across statements like: There are infinitely many reals. I know that reals are non-...
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2answers
59 views

Backwards Zeno's Paradox [on hold]

As I understand it, correct me if I am wrong, walking a finite distance will not take an infinite amount of time, because although you have to travel an infinite number of finite distances, the time ...
2
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3answers
90 views

Suppose $X$ is infinite and $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous

Suppose that $X$ is infinite and that $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous. My attempt: Let $|A|=n$. We will prove by induction on n. It's clear that the the ...
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3answers
52 views

A wrong definition of infinity and a doubt about its modification

In this answer, @Hyperplane defined an infinite set as follows: Well a set $M$ is infinite if for every finite subset $U \subsetneq M$ there exists a $x \in M, x\notin U$. Consequently, you will be ...
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1answer
52 views

What is $(-1)^\infty$, or $(-1)^x$ as the limit of $x$ approaches infinity?

I typed it into Symbolab calculator and it said that the solution diverges, though in real numbers the solution would never go past 1. Why is this?
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1answer
50 views

Does there exist an interval on the reals, however small, in which every number is irrational? [duplicate]

Since the irrationals are uncountably infinite, I've always imagined them "filling the space" between the rational numbers, but does that make sense? Or is it the case that any two irrational numbers ...
0
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1answer
38 views

Is infinity part of the boundary of R^n?

I am having a set $S=\{(x,y) \in \mathbb{R}^2 | x\in(-1,1), y\in(0,\infty) \} $. Does $\bar S$\S include the set of points for which $y\to\infty$; figuratively speaking, does it include the "boundary" ...
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3answers
43 views

Finding sum of a geometric series

I am asked to find the summation of $1/3^n$ from $n=5$ to infinity. I have done the calculation: $1/(1-r)$, for $r=1/3$, and received $1.5$. As this summation starts from $5$, I subtracted $3^0, 3^-1,...
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1answer
47 views

what does it mean ' infinity comes in different sizes'? [duplicate]

What does mean when mathematicians say infinity comes in different sizes ?
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2answers
23 views

How can we prove $\sum\limits_{k=0}^{\infty}(m+k+1)^nx^k=\sum\limits_{q=0}^{n}\binom{n}{q}m^{n-q}\frac{A_{q}(x)}{(1-x)^{q+1}}$?

If $$\sum\limits_{k=0}^{\infty}(k+1)^nx^k=\frac{A_{n}(x)}{(1-x)^{n+1}}$$ so $$\sum\limits_{k=0}^{\infty}(m+k+1)^nx^k=\sum\limits_{q=0}^{n}\binom{n}{q}m^{n-q}\frac{A_{q}(x)}{(1-x)^{q+1}}$$ How can we ...
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0answers
80 views

Is this proof of $0 = \infty$ just a mathematical joke? [duplicate]

Is this "proof" just a mathematical joke or might there be some deeper truth in it, eventhough the theorem is obviously false? Definition: Let a regular $n$-gon be a geometric figure $f^d_n$ that ...
0
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1answer
19 views

F distribution with the denominator degrees of freedom infinitely large

Definition Let $W$ be a chi-squared random variable with $m$ degrees of freedom and let $V$ be a chi-squared random variable with $n$ degrees of freedom, where $W$ and $V$ are independently ...
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2answers
86 views

How can i put a log inside an infinite sum? [closed]

iI would like to turn this $$ e^z-1 = \sum^\infty_1 \frac {z^n}{n!} $$ into this $$ \text{something} =\sum^\infty_1 \frac {z^n}{\log_e(n!)} $$ Is this at all possible? thank you very much in ...
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2answers
70 views

If $f:A\rightarrow A$ is injective but not surjective then $A$ is infinite.

I want to follow from something that peano axioms state about the successor function $s:\Bbb{N}\rightarrow\Bbb{N}$. Axiomatically $s$ is injective and not surjective. Now, $\Bbb{N}$ is infinite, just ...
1
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1answer
86 views

How to mathematically/logically prove that everything in our “real” world is finite? [closed]

Ok. This sounds like a kid question made when she/he first heard about "infinite". (But is that even bad?) So, I'll give a lecture about (cardinals) infinite numbers for some 12-15 yo kids and I wish ...
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2answers
37 views

Infinite size is actually number density? [closed]

Am I wrong in thinking that saying one infinity is larger than another is slightly disingenuous? My argument being the infinity is actually more dense than 'larger'. Obviously, we are already thinking ...
1
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1answer
63 views

Determining whether elements map to $\aleph_0$

Consider the infinite sequences $$A=\left\{\begin{array}{lcr} 1,\\2,2,\\3,3,3,\\4,4,4,4,\\\vdots\end{array}\right\}\qquad B=\left\{\begin{array}{lcr} 1,\\2,3,\\3,4,5,\\4,5,6,7,\\\vdots\end{array}\...
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2answers
39 views

For what values of $a$ and $b$ does $\lim_{x\rightarrow \infty}(\sqrt{x^2+x+1}-ax-b)=1$?

I have a question about limits tending to infinity. I need to find the constants $a$ and $b$ for which this limit takes the value 1. Please, help! Thank you! $$\lim_{x\rightarrow \infty}(\sqrt{x^2+...
3
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2answers
109 views

Can $\mathbb R$ be written as $(-\infty , \infty)$?

I was thinking about if $\mathbb R$ could be written as $(-\infty , \infty)$. I'm not sure if it's okay, because I've read somewhere (I can't remember where) that $(-\infty , \infty)$ declares ...
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0answers
39 views

Can the “symmetric algebra” over $\mathbb R^n$ be defined from an infinite-dimensional exterior algebra?

https://en.wikipedia.org/wiki/Symmetric_algebra If I understand that article correctly, the symmetric algebra $S(\mathbb R^n)$ is (isomorphic to) the algebra of polynomials with $n$ variables. As a ...
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1answer
38 views

How can I find a finite closed form for $\phi (\lambda)$?

How can I find a finite closed form for $\phi (\lambda)$, which that $$\lim_{\lambda \to \infty} \frac{\sum_{n=1}^{\lambda}2^{n^2}}{\phi(\lambda)}=1~~~~~~~~\lambda \in \mathbb{Z^{+}}$$ Is this ...
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3answers
95 views

Explain the odd/even inequality in the heights of numbers under the Collatz $(3x+1)/2$ transformation?

My kid asked me a question and I'm finding it hard to answer: if every number under repeated application of the Collatz transformation1 eventually reaches $1$, then it must be true for both even and ...
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1answer
66 views

Can there be a manifold with uncountable genus? Can they be compact?

Just some random thoughts I had after coming across this paper on Riemann Surfaces of Infinite Genus: [http://www.math.ubc.ca/~feldman/papers/allriem.pdf] I'm reasonable convinced that $\mathbb{R}^n$ ...
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2answers
119 views

Is the cardinality of the real numbers $\mathfrak{c}$ a real number?

This may be a stupid question, but I was learning some set and group theory and it just made me think. Clearly the continuum is an infinite quantity $\mathfrak{c}$, but the set of all reals is also ...
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3answers
364 views

How can we identify a set is infinite when proving Bolzano–Weierstrass theorem?

I'm proving the Bolzano–Weierstrass theorem. It says $$ \text{If } A \text{ : bounded and infinite set}, \text{ then } A \text{ has at least one limit point.} $$ So the proof goes like the following....
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3answers
72 views

Adding always the half of the previous number to a number beginning with 1…

This question blows my mind. Let's say we start at 1 and keep adding always 0.5 * the previous number. 1 + 0.5 + 0.25 + 0.125 + ..... The question is: will it reach infinity? I really don't know, ...
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1answer
38 views

Infinite reward in infinite steps [closed]

Let's say we have a problem where we get ∞ reward(can be substituted for anything) in ∞ timesteps. We want to know how much reward we get after one. This leads to:$\frac \infty\infty$ = $\frac 1x$ ...
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1answer
25 views

How does $e$ arise in solving the differential equation $\frac{dG(s,t)}{dt}=r(s-1)G(s,t)$?

I have the following equation $$\frac{dG(s,t)}{dt}=r(s-1)G(s,t)$$ which is supposed to yield the following when integrating from 0 to $t$ over $t$ (on both sides); $$G(s,t)=e^{r(s-1)t}G(0,t)$$ Any ...
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0answers
70 views

What's a Π proposition?

I'm trying to understand Indescribable Cardinals, but the Wiki article requires understanding what a Πm proposition is. I've tried Googling all the linked articles (...
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4answers
690 views

Adding infinite and finite numbers: why doesn't 0=1?

Okay, so, $$\infty + 1 = \infty$$ subtract infinity from both sides. $$1=0$$ At first I thought, duh, $\infty \neq \infty+1$, but, now, I'm just more confused because my brother rephrased it in ...
2
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1answer
55 views

Area of a line - $\infty$ or $0$

So today read something about two and three dimensional co-ordinate system and also about infinity and something came to me that I have been since pondering on. So my question is, what is the area of ...
2
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3answers
94 views

Does countable induction over $\mathbb N$ require the axiom of infinity?

Consider ordinary induction over $\mathbb N$: Proving $P(0)$ Assume $P(k)$ being true for some natural $k$ Prove $P(k+1)$ Does this require the existence of $\mathbb N$ and hence the ...
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4answers
106 views

How to prove that $\lim_{x \to \infty}\int_{x}^{x+1}\frac{{t^2}+1}{{t^2}+{20t}+8}dt=1$?

What I did to prove that $\lim_{x \to \infty}\int_{x}^{x+1}\frac{{t^2}+1}{{t^2}+{20t}+8}dt=1$ is: Because in the integrals ${x \to \infty}$, then the constant & ${constant*t}$ will not impact the ...
3
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2answers
73 views

why the integral $\int_{n=0}^{\infty} \frac{dx}{x^8 + \sqrt{x}}$ is converge?

How to show that this integral $\int_{n=0}^{\infty} \frac{dx}{x^8 + \sqrt{x}}$ is divergent? I try using $g(x)=\frac{1}{x^8}$ and then $\lim_{x \to {\infty}} \frac{\frac{dx}{x^8 + \sqrt{x}}}{\frac{1}{...
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3answers
44 views

Justifying the difference between countable and uncountable

Recently, I had sets $A$ and $B$, and needed to prove that there existed some element of $A$ that was not in $B$. I did this by showing that $A$ was uncountably infinite and $B$ was countably infinite,...
1
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1answer
65 views

Proof that it is unsolvable whether there's an infinity between countable and uncountable? [duplicate]

I have recently watched a video by "Undefined Behavior", explaining countable and uncountable infinities, and showing why uncountable infinity is larger than countable infinity. He then stated that a ...
7
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1answer
248 views

Analogue of complex infinity for quaternions

The complex infinity is defined as a pole on the Riemann sphere, which is the result of the 1-point compactification of the complex plane. Considering that quaternions are an extension of complex ...