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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2answers
50 views

Sizes of infinities

How is it that some infinities are bigger than other infinities and also If I have an infinite amount of apples and an infinite amount of planet earths then which will have the greater mass? (My ...
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4answers
55 views

Special limit: $1^\infty$

This is my limit, $\lim\limits_{n \to \infty } (1+\frac{1}{n^2})^{3n^2+4}$. If I place the infinity instead of $n$, I get $1^\infty$. I know that this is a special limit, but how I need to ...
-4
votes
1answer
42 views

Proving Limit is a sequence of positive numbers

$$\lim_{n \rightarrow \infty}{ a_n=0}{\implies \lim_{n \rightarrow \infty}{\frac{1}{a_n}=\infty}}$$} Assuming this sequence is of positive numbers, can anyone give any tips how to start proving this? ...
0
votes
1answer
32 views

Limit logarithm $\lim_{n \to \infty}(4n-6)[\ln(2n+5)-\ln(2n-7)]$.

please any advice for this limit? $$\lim_{n \to \infty}(4n-6)[\ln(2n+5)-\ln(2n-7)]$$ Thanks for any advice.
1
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1answer
30 views

Does the differentiation of countable and uncountable infinities play any role in calculus?

Calculus uses the concept of infinity a lot. I have never seen the type of infinity to make any difference in calculus. Are there any ideas in calculus that care what flavor of infinity is to be used? ...
3
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0answers
43 views

Kempner's series vs harmonic series (convergent vs divergent series via exclusion)

$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots$$ Okay so we all know the harmonic series is divergent right? But apparently when you remove all the terms that has a nine in it, ...
-4
votes
1answer
73 views

Infinite sum of cosine function [closed]

What does the following expression equal to $$\sum\limits_{n=1}^\infty \cos(n\cdot\theta)=\text{?}$$
1
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1answer
73 views

Interpretation of 2 proofs involving limits at infinity and mathematical induction

I have 2 exercises that I think are related to each other. I think they should be proved by mathematical induction. They are: prove that: limit of n which approaches infinity $(2^n / n!) = 0$ prove ...
0
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0answers
21 views

Simple types of infinites [duplicate]

I understand that some infinities are bigger than others and there are different types (I know of countable infinities), but are there other types of infinities and could you please explain them ...
0
votes
2answers
37 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 ...
2
votes
3answers
88 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
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1answer
62 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
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1answer
73 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
43 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: $\lim z \to \infty \left (z^2 +\sqrt{z^{4}+2z^{3}}-2\sqrt{z^{4}+z^{3}}\right )=\frac{-1}{4}$
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votes
1answer
56 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
85 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 ...
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
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0answers
32 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 ...
9
votes
4answers
146 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
vote
1answer
88 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 ...
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votes
2answers
67 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
39 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 ...
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2answers
110 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
55 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
58 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
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1answer
112 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
81 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
votes
0answers
33 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 ...
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1answer
58 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
286 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
16 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
95 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. ...
0
votes
1answer
19 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
187 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
76 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
35 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 = ...
3
votes
5answers
81 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
58 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} ...
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votes
1answer
94 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 $\sum^\infty_{n=1}n\rightarrow+\infty$?
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1answer
70 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
104 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 ...
-4
votes
1answer
58 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
24 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
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0answers
62 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
72 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
66 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 ...