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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0
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2answers
39 views

Question about evaluating infinite limit

I have to evaluate an positive infinite limit for $$\lim_{x\to\infty}{\sqrt{1+4x^6}\over 2-x^3}.$$ I did it my way by squaring the whole thing, which gets rid of the square root, then I just foil the ...
3
votes
3answers
131 views

Is $\omega_0-1$ infinite?

I have read in another answer Is infinity an odd or even number? that the $\omega_0$ is the "smallest infinity", but is $\omega_0-1$ not also infinite?
0
votes
1answer
69 views

Use fourier transform to solve second-order differential equation — an “easy” integral?

I have scoured the internet for a fully-explained solution to this problem but have found none: The problem asks to solve this differential equation for $y(t)$ using Fourier Transforms, and then ...
2
votes
1answer
75 views

How many objects are in $\mathbf{Set}$? [closed]

... or does this question even make sense, considering the object "collection" of $\mathbf{Set}$ is a proper class rather than a set?
2
votes
1answer
44 views

Countable additivity and $P(\Omega)=1$

The first axiom of probability states that $P(\Omega)=1$. On the other hand, we have that $$P\left(\bigcup_{i = 1}^\infty E_i\right) = \sum_{i=1}^\infty P(E_i)$$ where $E_i$ are pairwise disjoint ...
3
votes
4answers
54 views

Limits without L'Hopitals Rule ( as I calculate it?)

Prove that: $\lim z \to \infty \left (z^2 +\sqrt{z^{4}+2z^{3}}-2\sqrt{z^{4}+z^{3}}\right )=\frac{-1}{4}$
-2
votes
1answer
66 views

Difference between arbitrarily large and infinite in terms of countableness

By the Cantor slash argument, as explained for example here (at about 4:00), a new real number can always be generated out of any list of real number decimal expansions by taking the digits along the ...
0
votes
1answer
173 views

Sum of positive infinity and negative infinity

Consider the following function of $\tau$: $$ h(\tau) := C_1 \ln\left(1-\frac{a}{\tau}\right) - C_2 \ln\left(1-\frac{b}{\tau}\right), $$ where $a > b>0$ and $C_2=\ln(1-b)<C_1=\ln(1-a)<0$....
3
votes
3answers
68 views

How Does the ($\sqrt{x^2+x}+x)$ Equal $(\sqrt{x^2}+x)$ When Calculating The Limit of Infinity?

I am asking this because of the following question: What is the Limit of positive infinity for the equation $\frac{1}{\sqrt{x^2+x}+x}$? The following steps are done to get the answer, which is 2. ...
1
vote
0answers
35 views

Probability of termination of random teleportation

In Minecraft, with mods, there's a liquid called Resonant Ender, which if you touch it, teleports you randomly up to 8 blocks on both the north-south and east-west axes. Consider an infinite sea of ...
10
votes
4answers
168 views

Does it make sense to define $ \aleph_{\infty}=\lim\limits_{n\to\infty}\aleph_n $? Is its cardinality “infinitely infinite”?

I recently read a book about infinity, which introduced the basic notions of different kinds of infinity. I'm a total layman concerning this topic, and one question fascinated me: Can we, in some ...
1
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1answer
94 views

Mathematical descriptions of physical space

Bear with me as I'm a philosophy (not math) student. First some philosophical background, and then the math question. One philosophical view is that physical space is composed of infinitely many ...
-2
votes
2answers
71 views

Finding the limit without L'Hopital's rule´s [closed]

I can not solve this limit $\lim_{ x \to \infty} \sqrt{x^{3}}(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x})$
3
votes
3answers
52 views

What is limit of $\displaystyle{\lim_{x \to -\infty}\sqrt[3]{x^{2}+\sqrt[3]{27x^{4}+\sqrt[3]{x^{2}}}}-\sqrt[3]{x^{2}}}$ whithout L´hopital.

$$\lim_{x \to -\infty}\sqrt[3]{x^{2}+\sqrt[3]{27x^{4}+\sqrt[3]{x^{2}}}}-\sqrt[3]{x^{2}}$$ Thanks in advance.
3
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2answers
46 views

Are more than 50% of integers below a positive x?

If I have a positive x, are there more integers below x or above x? I was discussing this ...
-1
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2answers
119 views

How can we think about infinity? [closed]

Consider the following abstraction. We have a skyscraper with an infinite number of floors. The first floor contains the first type of infinity $\aleph_0$. So I guess finite numbers can live in the ...
2
votes
1answer
65 views

Approaching infinities

How big of an infinity can we approach? At first I naively thought about: $$ \lim_{i \to \aleph_0}\aleph_i $$ However, this approaches $\aleph_{\aleph_0}$ which is $\aleph_{\omega}$ Let's define: $$ ...
0
votes
1answer
59 views

Calculus, limits: Can someone explain to me why $-\frac{\sqrt{2+x^2}}{2x}$ is equal to $-\frac{1}{2}$ when $x$ approaches $\infty$?

I'm reviewing for my midterm in 3 hours and just came across this practice question/solution and don't understand it. Thank you!
2
votes
1answer
116 views

Limits of transfinite numbers

Does it makes sense to talk about: $$ \lim_{i\to \aleph_0} \aleph_i $$ What type of infinity does it approach? Maybe finding a limit of that doesn't make sense. What about $\aleph_{\aleph_0}$? ...
0
votes
1answer
111 views

Proving that a regular polygon with infinite sides is a circle by using limits on the formula $\frac{\pi}{n}(n-2)$

In childhood, when we were taught circles for the first time, our teacher always told us that a circle is like a polygon which has infinite sides. But how to prove it? A regular polygon's interior ...
0
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0answers
38 views

$\Sigma^{\infty}_{n=1}(-1)^n[ \frac{\pi sin(n\theta)}{n}-\frac{2cos(n\theta)}{n^2}]=0$

Is $\Sigma^{\infty}_{n=1}(-1)^n[ \frac{\pi sin(n\theta)}{n}-\frac{2cos(n\theta)}{n^2}]=0$ true? This is came out from my Fourier series computation, according to the answer, that sum should be zero ...
-4
votes
1answer
74 views

infinity, countable infinity and uncountable infinity [closed]

I'm asking about the concept of infinity. How do you define the concept of countable infinity and uncountable infinity? Please explain these there concepts; infinity, countable infinity and ...
2
votes
6answers
336 views

What is $0 \times \infty$? [duplicate]

My question is - I know, $0\times anything=0$ and $anything \times \infty=\infty$. So,what is $0 \times \infty$? I suppose it's $0$ but why not $\infty$? If I say that area of an indefinitely long ...
1
vote
1answer
17 views

Get sum with limits

I have a sum $\frac{1}{2+a}$ which is only valid for $a \in (-\infty,-3) \cup (-1,\infty)$. To express the some correctly, should I show the sum for the entire interval? I guess I can say something ...
2
votes
3answers
201 views

infinite subset of an finite set?

Is it possible to have a set of infinite cardinality be a subset of a set with a finite cardinality? It sounds like there shouldn't be but there are some thing in math that sound counterproductive. ...
1
vote
1answer
25 views

Formal Way to Prove limit without operating on infinity?

What is the "formal" way of proving Limit as x approaches negative infinity of f(x) Where f(x) = sqrt(5-x) I know it's positive infinity but in order to get that I had to "operate" on infinity which ...
2
votes
2answers
53 views

Calculating $\lim_{x\to0^+}x-\frac{1}{x^3}$

$$\lim_{x\to0^+}x-\frac{1}{x^3}$$ The answer is $-\infty$. It's not very clear to me how was that concluded. You can't plug in the $0$ because you'd have $\frac{1}{0}$ which is indeterminate. But ...
1
vote
5answers
194 views

Find the limit of $x +\sqrt{x^2 + 8x}$ as $x\to-\infty$

$$\lim_{x\to -\infty} x +\sqrt{x^2 + 8x}$$ I multiplied it by the conjugate: $\frac{-8x}{x - \sqrt{{x^2} + 8x}}$ I can simplify further and get: $\frac{-8}{1-\sqrt{1+\frac{8}{x}}}$ I think there ...
3
votes
4answers
78 views

How do I calculate $\lim_{x\to+\infty}\sqrt{x+a}-\sqrt{x}$?

I've seen a handful of exercises like this: $$\lim_{x\to+\infty}(\sqrt{x+a}-\sqrt{x})$$ I've never worked with limits to infinity when there is some arbitrary number $a$. I am not given any details ...
1
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0answers
36 views

Show that $l_{2}(J)$ is Hilbert Space for Countably Infinite Set?

The inner product is \begin{equation*} \langle u, v \rangle = \sum\limits_{j \in J} u_{j} \overline{v_{j}} \end{equation*} where $u,v$ are vectors and $J$ is the countably infinite set $J = \mathbb{...
3
votes
5answers
82 views

Limit at infinity for sequence $ n^2x(1-x^2)^n$

I'm supposed to prove that this sequence goes to zero as n goes to infinity. $$\lim_{n\to \infty} {n^2x (1-x^2)^n}, \mathrm{where~} 0 \le x \le 1$$ I've been trying a few things (geometric formula, ...
2
votes
1answer
62 views

iterated sine function on different arguments

I want to evaluate the following: $\lim_{n\rightarrow \infty} \sqrt{n} \sin^{(n)}(2/\sqrt{n})$, where $\sin^{(n)}$ is the iterated sine function. I do know the proof for $\lim_{n\rightarrow \infty} \...
-4
votes
1answer
103 views

Why $\zeta(-1)=-\frac{1}{12}$ does not mean the sum from $1$ to infinity is $-\frac{1}{12}$ [duplicate]

Since $\zeta(-1)=\frac{1}{1^{-1}}+\frac{1}{2^{-1}}+\frac{1}{3^{-1}}+\cdots=-\frac{1}{12}$, why do we still say that $\sum^\infty_{n=1}n\rightarrow+\infty$?
0
votes
1answer
76 views

What is infinity added to itself a countably infinite number of times?

What is infinity added to itself a countably infinite number of times? Intuitively, it seems to me that $$\sum_{n=1}^\infty \infty = \infty \cdot \infty = \infty,$$ because $$ \sum_{k=1}^n \infty = n ...
2
votes
2answers
49 views

Is there a categorizaiton system for null quantities?

Many of us are familiar with the transfinite numbers as representing different levels of infinity. I was wondering if there were a similar system for categorizing null quantities? My motivation for ...
2
votes
3answers
119 views

Limits at Infinity proof

The problem is prove the limit using definition 6, $$\lim_{x\rightarrow-3} \frac{1}{(x+3)^4} = \infty$$ The book gives definition 6 as: Let $f$ be a function defined on some open interval that ...
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votes
1answer
76 views

Which one is bigger, infinity sign(∞) or aleph number? [closed]

the infinity sign(∞) is often used casually but it is very abstract concept and ill-defined... when there are 'infinite' natural numbers and aleph-zero is cardinality of a set of natural numbers.. is ∞...
0
votes
1answer
26 views

How to calculate convolution with logarithm numerically?

I'm trying to compute an optimisation problem, which has a cost function involving $$I=\int_0^1\log|x-y|\rho(y)dy$$ where $x\in[0,1]$ and $\rho$ is a probability density. Eventually, I will want to ...
0
votes
0answers
65 views

Probability of Unions to infinity

I had a question about the probability of unions to infinity. 1) Everyone in a group of $N > 3$ people writes their name on a slip of paper and drops the slips into an urn. Then, one at a time (...
0
votes
1answer
81 views

Find the value of $ \sum_{n=1}^{\infty} \frac{n^3}{3^n} $ [duplicate]

friends. The question is: Find the value of $ \sum_{n=1}^{\infty} \frac{n^3}{3^n} $. I know this sum converges and that it's value is $ \frac{33}{8} $, however, I can't seem to find it. I've tried ...
2
votes
1answer
82 views

P-adic numbers and infinity? Does infinity as a limit exist for p-adics?

I don't think I understand how p-adic numbers relate to the usual concept of infinity. The wiki page and various sources on the internet did not help. Let's see for example the 10-adic counterparts ...
0
votes
3answers
116 views

Is $\lim\limits_{x,y\to-\infty}\frac{\sqrt x\sqrt y}{\sqrt{xy}}=1$?

WolframAlpha is suggesting (judging by the plot given) that the limit is actually $-1$. I would think the following manipulations would be okay to conclude that is the opposite. $$\lim_{x,y\to-\infty}...
7
votes
7answers
540 views

What is $2!!!!!!!!!!!!!!!!!!!!$… (up to?

A few days back a question came to my mind What is the value of $2!!!!!!!!!!!!!!!!....$ (up to infinity)? I feel it is 2, but one of my friends said that we can't say that for infinity. I know ...
4
votes
4answers
116 views

How is $0\cdot\infty= -1$?

It is known that the product of slopes of two perpendicular lines is equal to $-1$ ($m_1*m_2=-1$ for $m_1$ and $m_2$ being the slopes of the perpendicular lines $l_1$ and $l_2$). The slope of $x$-axis ...
1
vote
2answers
110 views

Infinitely Counting Real Numbers [duplicate]

I would like to show an idea on how to make real numbers infinitely countable. It is quite simple, too simple for me to believe it has been overlooked. So my question is, what have I overlooked? So ...
6
votes
3answers
260 views

The numerical relation of the sum of two divergence series

For these two series: $1 + 2 + 3 + 4 + 5 +...$ $2 + 4 + 6 + 8 + 10 +...$ For each of the two series, since these numbers progress with no end, and the sum increases, it certainly cannot be finite. ...
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votes
2answers
360 views

How is the set of natural numbers countably infinite. [closed]

Here is my question TLDR. How can the natural number system be considered "countably infinite", when it has subsets that are infinite within itself. Isn't a countably infinite set one that contains ...
1
vote
3answers
526 views

The cardinality of the even numbers is half of the cardinality of the natural numbers? [duplicate]

The following match-up makes it clear that the set of even integers and the set of positive integers have the same cardinality(size) since it establishes a one-to-one correspondence between them: ...
3
votes
2answers
52 views

Limit of a trigonometric rational expression

How to evaluate the limit of this expression? $$\lim_{x\to\infty} \frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}$$ I managed to simplify the denominator into a sinus form by the ...
1
vote
1answer
99 views

Why in Hilbert's Hotel paradox can you not simply put the new guests at room $n+1$?

I understand how in the infinite hotel 'paradox' moving every person in room $n$ to room $n+1$, and then putting the new quests in room $1$, generates a new space in the countable, but infinite, set. ...