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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Once we've constructed N, do we need again the axiom of infinity to conclude the set of the primes is infinite?

When completing the proof (unfortunately, at the same time presenting it as Euclid's and performing it ad absurdum) of the infinitude of prime numbers, my algebra professor stated "...and thus the set ...
2
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3answers
78 views

How do I compute $\sum_{k=1}^{\infty} k \cdot p^k$ [duplicate]

I have no idea how to compute this infinite sum. It seems to pass the convergence test. It even seems to be equal to $\frac{p}{(1-p)^2}$, but I cannot prove it. Any insightful piece of advice will be ...
2
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1answer
69 views

Infinite lines and points

Ok, lately I have been thinking a lot about one idea that has been bothering me since first I learned about lines and points. I understood that: A line has no thickness, is straight and it is ...
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1answer
348 views

Why does this mathematics Professor believe -1/12 is a mysterious value related to 'infinity'? [duplicate]

In the following YouTube video, a maths Professor talks about -1/12 and its mysterious relationship with infinity: "Why -1/12 is a gold nugget" He says, about the sum of the powers of natural ...
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1answer
35 views

Can infinity be represented as a recursive function?

Could infinty be represented as a recursive function like: The function f takes any number x as parameter and returns f(x+1), resulting in an endless recursive call, each call incrementing x by 1. ...
2
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1answer
88 views

What does this product converges to?

Let $p\in[0,1]$. I'm interested in computing $$\lim_{n\to\infty}\prod_{i=1}^n(1-p^i)$$ Any thoughts? EDIT: As Kibble mentioned, this is the Euler function. Also from Kibble: a simple upper ...
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6answers
54 views

how to find an infinity limit in a fraction

I don't understand how to find the limits of this expression when $x\to\infty$ and $x\to-\infty$: $$\left(\frac{3e^{2x}+8e^x-3}{1+e^x}\right)$$ I've searched for hours. How to compute these limits?
3
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3answers
98 views

$f(x)$ has a limit, prove that $\sqrt{f(x)}$ has a limit

The question is : Let $f$ be a positive function defined on an interval $[a,\infty)$, such that $\lim\limits_{x\to\infty} f(x)=0$. Prove that $\lim\limits_{x\to\infty} \sqrt{f(x)}=0$. If $f(x)$ ...
2
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1answer
30 views

Proving a limit on a general function

The question is : Not sure how to even start, i know that |F(x) - L| < Epsilon so if F(x) = L then the function is liner on L, Don't know how to prove its an integer. Thanks in advance !
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2answers
39 views

Solve using the limit's definition

The question is : I am stuck here : $$| \sqrt{x+1} - \sqrt{x} -1 | / \sqrt{x} + 1$$ i know that the numerator is negetive so i must change it in order to delete the absolute value yet i still ...
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2answers
31 views

Finding a very close epsilon

This is the Question : Given the following limits, find $M>0$ such that for every $x>M$ the expression are $\frac13$-close to their limit (in other words, find $M>0$ s.t. for every ...
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1answer
113 views

Proof that √x does not tend to a limit as x approaches infinity

I am wanting to prove from the definition of a limit ( $∀ε>0 ∃K>0:∀x>K, |f(x)−l|$<ε) that $√x $ does not tend to a limit as $x$ approaches infinity. So far I have tried to find a value of ...
1
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1answer
89 views

limit as x approaches zero from left and right equals positive and negative infinity

How can we prove \begin{align} \lim_{x\to 0^+}\frac1x &= +\infty, \\ \lim_{x\to 0^-} \frac1x &= -\infty \end{align} This seems really simple but I'm having trouble starting it
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2answers
78 views

Infinity In Math - The Nick Lim Proposal [closed]

So infinity is clearly a very strange, concept. So I have the following proposal (the nick lim proposal) which can not be solved ( at least to my current knowledge, hopefully you can shed some light ...
3
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0answers
37 views

The behavior of the 3D wave equation close to the origin

The general solution to the three dimensional wave equation is \begin{equation}u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r} \end{equation} where $F$ and $G$ are arbitrary functions. I want to ...
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3answers
49 views

What is the limit of the following sequence? $\lim_{n\to\infty} 8^\frac{n+1}{3n+2}$

What is the limit of the following sequence? $$\lim_{n\to\infty} 8^\frac{n+1}{3n+2}$$ I substitute infinity in $n$ and I get infinity + 1 = infinity, 3*infinity+2 = infinity. Infinity over infinity = ...
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2answers
51 views

Why is the limit of this function tending to 1?

$$ \lim_{x\to \infty} \left(\frac{1}{(x^2+x)\left(\ln\frac{x+1}{x}\right)^2}\right) $$ I know the answer is 1, but why does it tend to 1? Can you manipulate the function and the "$\ln$" to make it ...
1
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1answer
44 views

infinity over infinity and zero multiplied infinity in a calculation which gives (correctly) 1

I hope you don't put as a duplicate my question as it is thought for the specific case I am going to show you now, which is about a calculation. I read about the topic of infinity over infinity, ...
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2answers
718 views

Uncountable vs Countable Infinity

My friend and I were discussing infinity and stuff about it and ran into some disagreements regarding countable and uncountable infinity. As far as I understand, the list of all natural numbers is ...
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0answers
60 views

Definition of the rank of infinite matrix

How is the rank of an infinite matrix defined? Is it the same as in the finite case, i.e. the number of elements in a basis for some matrix? How are the dimensionalities of the column and null ...
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0answers
23 views

Is it formally correct to take the limit of infinitely many terms?

Let us take the most basic example: $\lim_{n\to\infty} \frac{n!}{n^n}$ $\frac{n!}{n^n} = \frac{n(n-1)...1}{n \cdot n...n} = \frac{n}{n} \cdot \frac{n-1}{n} \cdot [...] \cdot \frac{1}{n} \leq ...
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5answers
100 views

Why can't $\int_{-1}^1{\frac{dx}{x}}$ be evaluated?

So I just saw this question on brilliant.org and found that many people argued over whether this could be evaluated. My answer was 0 because it is an odd function, but others argued that because of ...
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2answers
73 views

How to write down proof that if $\lim_{x\to \infty}f(x)=\alpha$ then $\lim_{x\to \infty}f'(x)=0$?

Let $a, \alpha \in \Bbb{R}$; let $f: (a,+\infty)\to \mathbb{R}$ be differentiable; let $\lim_{x\to \infty}f(x)=\alpha$; let $\beta := \lim_{x\to \infty}f'(x)$. I want to show that $\beta = 0$. Now, ...
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2answers
22 views

Evaluation of this limit

I am looking in a calculator how to evaluate the limit as x approaches infinity for the following function: $\sqrt{x} -\sqrt{x-1}$. In the evaluation the calculator applies the following algebraic ...
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2answers
55 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
60 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 ...
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1answer
47 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
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1answer
39 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.
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1answer
34 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? ...
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0answers
49 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, ...
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1answer
75 views

Infinite sum of cosine function [closed]

What does the following expression equal to $$\sum\limits_{n=1}^\infty \cos(n\cdot\theta)=\text{?}$$
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1answer
81 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 ...
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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 ...
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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 ...
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3answers
92 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?
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1answer
68 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
74 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
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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 ...
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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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1answer
63 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 ...
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1answer
164 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 ...
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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. ...
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0answers
34 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 ...
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4answers
165 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 ...
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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 ...
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2answers
69 views

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

I can not solve this limit $\lim_{ x \to \infty} \sqrt{x^{3}}(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x})$
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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.
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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 ...
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2answers
117 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
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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: $$ ...