Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.

learn more… | top users | synonyms

2
votes
1answer
38 views

How can we compare a number with inifinity?

we know that $a<\infty\space\space \forall a\in\mathbb{R}$ but why? how can oder in $\mathbb{R}$ include this concept?
-2
votes
0answers
42 views

Extending our number system to include infinities [on hold]

Instead of writing infinity using the infinity symbol, could we write such numbers as: |$\mathbb Z$| (size of the set of integer numbers) |$\mathbb R$| (size of the set of real numbers) Then ...
1
vote
2answers
68 views

Is $\lim_{n \to \infty}\frac{a}{\frac{b}{n}}$ equal to $\infty$ or undefined?

Where $a$ and $b$ are constants. I can think of it two different ways. First is that as $n$ goes to infinity, $\frac{b}{n}$ goes to $0$, so that we end up with $\lim_{n \to \infty}\frac{a}{\frac{b}{...
2
votes
3answers
42 views

What happens when $r \to \infty$? Will it be a line? (partial circle)

Let $a$ be a arc of particle circles, which is constant. What happens when $r \to \infty$? Will it be a line? Radius of partial circle : $r$, Arc of partial circle : $a$ and constant, For $r=r_0$ ...
1
vote
1answer
21 views

Limit of a function with exponential function and two parameters tending to infinity

I need some help with calculation of limits. I have a function $n(e^{it/\sqrt{m+n}}-1) + m(e^{-it/\sqrt{m+n}}-1) + \frac{m-n}{\sqrt{m+n}}it$ The solution says this converges to $-\frac{1}{2}t^2 \...
3
votes
1answer
95 views

What is the derivative of $x^{x^{x^{x^{.^{.^{.}}}}}}$ [duplicate]

Here is my attempt: Substituting y for infinite x powers: $$x^{x^{x^{x^{.^{.^{.}}}}}}=y → x^y=y $$ Giving: $$x=y^{\frac{1}{y}}$$ Take natural logs & differentiate with respect to $y$: $$ln(x)=...
0
votes
0answers
36 views

D. F. Wallace's “Everything and more” $\S$7b : Cantor transfinite derivation from $P^{(n)}$

On $\S$7b of David Foster Wallace's book "Everything and more", the author explains how Cantor derived the concept of transfinite numbers from P, a second-species infinite point-set. "$P'$, can be "...
2
votes
4answers
93 views

Calculus Improper Integral Convergence; Which is right: Limits or Areas?

Could someone please explain to me the following doubt I have on improper integral: $$\int_{-\infty}^{\infty} \frac{1}{x} \ \mathrm{ dx}$$ I still think that since integrals signify areas that this ...
2
votes
2answers
41 views

Set of Natural Numbers, Choice, Computer Programs

I am aware that the following question/discussion is ill-defined, and perhaps has more to do with philosophy than mathematics. My brother and I were talking and the topic came to “picking a random ...
0
votes
0answers
42 views

Infinite sums and squaring the plane, or sort of

This question is regarding two algorithms for squaring/almost squaring the plane. the Henles' method of squaring the plane. pdf here my method of tiling $n^2$ squares. I worked out* a simple gap-...
-3
votes
1answer
62 views

1/$\aleph$ and probability

what is the significance of 1 divided by the various alephs ($\aleph$), the sets of reals, rationals, and so on. I was thinking about the probability of selecting a random real or integer number. I ...
0
votes
0answers
21 views

Random walk on a segment with infinite time

Given a point particle on a segment $L$ of length $1$, $(L=[0,1])$, assume the particle moving randomly in such a way: $p_{(k+1)}=p_k+\delta_k$ where $p_{k+1}$ is the position on the segment at time $...
2
votes
1answer
65 views

A Puzzle on Infinity: How to guess the color of hats? [duplicate]

Infinitely many (i.e. $\omega$ - many) people each have either a white hat or black hat on their heads. Each person can see everyone's hats except their own. Each person simultaneously announces a ...
1
vote
1answer
38 views

Inverse of a matrix with main diagonal elements approaching infinity

Let $A$ be a invertible, symmetric, positive definite $p \times p$ covariance matrix with main diagonal elements $a_{ii},~i = 1,~\ldots,~p$. If all main diagonal elements would approach $\infty$, ...
43
votes
16answers
3k views

Different sizes of infinity

Correct me if I'm wrong, but this is what they taught us in precal: $$\lim_{x\rightarrow\infty}x=\infty$$ $$\lim_{x\rightarrow\infty}x^{2}=\infty$$ But, we also know that $n^{2}>n$ if $n\notin [0,1]...
2
votes
1answer
103 views

What's so different about limits compared to infinitesimals?

If you find the limit is 2 for a given function, wouldn't this be the same as $2 + \epsilon$ with $\epsilon$ being a negligible value? This different way of defining limit-like behavior seems rigorous ...
0
votes
3answers
107 views

How $\infty=\infty$.

If we contruct two strainght lines as shown: Then join them such that to complete a triangle. It is taught that we can find infinity points on straight line. So there are infinity points on $DE$ ...
2
votes
1answer
68 views

Topology and Borel sets of extended real line

Let $\mathcal{B}_{X}$ denote the Borel $\sigma$-algebra on $X$. I'm reading a book on real analysis by Folland and he defines $$\mathcal{B}_{\overline{\mathbb{R}}} = \{ E \mid E \cap \mathbb{R} \in \...
0
votes
1answer
51 views

What are the limit points of $A_n=[n,\infty)$ in a metric space? Is $A_n$ closed?

$A_n=[n,\infty)$ in $\mathbb{R}$ with a Euclidean metric. A set is closed if it contains all its limit points. A limit point is a point whose neighborhood contains a point in the set. I'm not sure ...
0
votes
2answers
34 views

Infinite differentiability with a removable discontinuity?

I'm still a beginner with calculus. But this puzzled me. Let's say you had $f(x) = \frac{x^2-1}{x+1}$. It's discontinuous at one point. If you took the derivative infinitely many times, would the ...
2
votes
1answer
53 views

Is this sequence going to infinity, and how do we know that?

$a+\dfrac {a+\dfrac {a+\dfrac {a+\dfrac {:} {b}} {b}} {b}} {b}=?$ I've tried letting $\quad a+\dfrac {a+\dfrac {a+\dfrac {:} {b}} {b}} {b}=K$ Which makes the equation: $a+\dfrac {K} {b}=K$ $\quad$ ...
2
votes
1answer
88 views

Proof of Lemma 8.2.3 in Terence Tao Analysis 1 book

$\textbf{Lemma 8.2.3 }$ Let X be a countable set, and let $f:X \rightarrow R$ be a function. Then the series $ \sum_{x \in X} f(x)$ is absolutely convergent if and only if $$ sup \left\{ \sum_{x \...
4
votes
5answers
154 views

Why can't we just say that $\infty-\infty$ equals zero?

Let be $\lim\limits_{x\to \infty}x=A$ and $\lim\limits_{y\to \infty}y=B$. Can be $A-B=0$? If the answer is "no" , why? And my other example: $\displaystyle\int_{-\infty}^{\infty} \dfrac{x}{...
0
votes
3answers
62 views

Where did $\sqrt{x^2/x^2}$ come from in $\lim_{x \to -\infty}\frac{x+1}{\sqrt{x^2}} = \lim_{x \to -\infty}\frac{-1-1/x}{\sqrt{x^2/x^2}} = -1$?

I'm reading a calculus book and I saw the following limit solution. $$ \lim_{x \to -\infty}\frac{x+1}{\sqrt{x^2}} = \lim_{x \to -\infty} \left(\frac{x+1}{\sqrt{x^2}} \cdot \frac{-1/x}{-1/x}\right) = ...
2
votes
1answer
53 views

What is meant by $\lim_{x\to \infty^+}$

I am familiar on how limits work and such. For example, look at the following limit: $$\lim_{x\to 5^+} \frac{-x^2+5x}{5-x} = 5$$ It is saying that, as $x$ approaches $5$ from the right, the equation ...
0
votes
1answer
44 views

Compute H-infinity norm in Matlab

Please can someone write a command in Matlab for calculating $H_{\infty}$ norm for the following system: $$\frac{d}{dt}z(t)=Az(t)+Bu(t)+Fw(t)$$ $$y(t)=Cz(t)+Du(t)$$ where $A$, $B$, $C$, $D$, and $F$ ...
2
votes
2answers
78 views

Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
0
votes
2answers
41 views

Can one non-cardinal infinity be greater than other non-cardinal infinity?

As far as I know, there are two different notions to the word "infinity" in Mathematics. First notion of infinity has to do with the cardinality of a set: if a set contains infinite number of ...
1
vote
1answer
83 views

Set theory with multiple countable infinities [closed]

In set theory, all sets that are countably infinite are generally considered to have the same size since there is a bijection between them. Has anyone tried formalising set theory in a way which ...
-3
votes
2answers
65 views

If $\dfrac{1}{\infty}=0$ then I can prove that $0 = 1$ [closed]

Given, $\dfrac{1}{\infty}=0$, then $1=0 \cdot \infty = 0$ (because $0$ times any number or values is $0$ and here that number is infinity). Which gives us $1=0$ i.e, $0=1$. Hence proved....
0
votes
1answer
34 views

Voronoi edges example

I have 4 line segments: 0 0 2 0 // 1st line segment 2 0 2 1 // 2nd line segment 2 1 0 1 0 1 0 0 and I wrote some CGAL code to print the Voronoi edges. However, <...
3
votes
5answers
888 views

How many points in a line segment?

My teacher said that in the circumference of circle there are infinite points. When I was learning more about circle, I came to this picture: My question is: When we unroll the circle, then the ...
0
votes
0answers
37 views

Is there an infinity smaller than countable? [duplicate]

In other words: is $\aleph_0$ the smallest infinity? Is it easy to prove?
-1
votes
2answers
77 views

Random Room changing in the Hilbert hotel. [closed]

Let's say you have a Hilbert's grand hotel full occupancy. Assign each occupant a new room select randomly without regard to whether the room is assigned to someone. i.e. empty rooms, multiple ...
4
votes
0answers
47 views

Flea on the coordinate system

We drop a flea on a point of the coordinate system(with integer coordinates). Due to the dimensions of the flea we can not see it. The flea jumps away every second by one unit (always in the same ...
0
votes
0answers
29 views

Complex variable limit at infinity

Is $\lim\limits_{z\to\infty} \frac{4z^2}{(z-1)^2}$, $z\in\mathbb{C}$, evaluated the same way as a real variable function limit? Or does one need to show separate cases for $x\to\infty$ and $y\to\infty$...
1
vote
1answer
46 views

Limit of $(-1/2)^n$ as $n$ approaches infinty

I tried plugging bigger and bigger $n$'s into my calculator and the result obviously approaches $0$ (albeit oscillating between positive and negative). So how do you prove that: $$\lim_{n \to \infty}...
1
vote
3answers
50 views

Arithmetic Operations with Infinities in Real Analysis

Infinity is not a number , thus we cannot perform the usual arithmetic operations that we do with real numbers This is the usual reason given when asked why we can't perform the usual arithmetic ...
5
votes
4answers
212 views

First year calculus student: why isn't the derivative the slope of a secant line with an infinitesimally small distance separating the points?

I'm having trouble with the limit approach to calculus ever since I heard about the infinitesimal definition. Maybe you can help me settle what's been bothering me this year. Looking at the limit ...
0
votes
0answers
41 views

Difficulty in understanding Cantor's diagonal argument

I recently found Cantor's diagonal argument in Wikipedia, which is a really neat proof that some infinities are bigger than others (mind blown!). But then I realized this leads to an apparent paradox ...
0
votes
0answers
17 views

Average of left and right limits | Signum function, Heaviside step function, and Grandi's series

This question probably already has an answer but usually involves stuff that's way over the top of my head so I'm hoping for a simple explanation. In Adams, R. A., & Essex, C. (7th edition) ...
0
votes
1answer
47 views

What is the formal definition of a limit at infinity?

I keep coming across two different kinds of answers to this question. The first definition: We say that $$\lim_{x\to \infty} f(x) = L$$ if the following condition is satisfied: for every number $\...
3
votes
1answer
31 views

How do I calculate this limit when two terms tend to infinity at similar rates

In a particular problem that I am currently trying to solve, I have the following expression (this is not the entire expression, I have included only the terms involving $a_1$ and $b_1$), $\lim_{(...
3
votes
1answer
32 views

Interval notation: infinity, -infinity in closed interval

I was watching a video stream a little bit ago and noticed on an equation without context that had the interval $\left[{-\infty, \infty}\right]$. This was preculiar to me as I've never seen the ...
1
vote
1answer
57 views

precise definition of a limit at infinity, application for limit at sin(x)

(a) Write down the first principles definition of the statement $\lim\limits_{x→∞} f(x) = L$. For this I have that for every $ε >0$, there is a corresponding number $N$, such that if $N>0$, ...
1
vote
2answers
71 views

did i use infinite wrong?

This algebra question is in Dutch and the original file van be found here: Question 19 Ill try to translate the important info needed to answer this question. $$s= \frac{(a+b)} { (ab)}$$ S= dpt ...
1
vote
2answers
42 views

Will a decreasing probability ever resolve favorably? [duplicate]

Let's say I start off with a 50/50 chance at winning the lottery. But I lose. Now my chance is only half as good, or 25%. I lose again. Now the chance is 12.5%. Same result. If this continues all ...
-1
votes
1answer
57 views

Find a **bijection** between two intervals

I am struggling with this question and was hoping somebody could help me, Thanks Find a bijection between the intervals $(-1,1)$ and $(0,4)$ where $X \in R$
2
votes
3answers
42 views

Division of segments into infinitely many parts.

Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2. If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested ...
0
votes
0answers
168 views

When adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?

I was watching Ramanujan: Making sense of 1+2+3+... = -1/12, where the presenter writes: (I tried to write this out in $\LaTeX$ but couldn't figure out how to do multi-column alignment without ...