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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integral $\int \frac{\infty(\frac{1}{x})}{x^3}dx$ [closed]

Is anyone familiar with the following notation: $\infty \left(\frac{1}{x} \right)$? I am solving the integral $\int \frac{\infty(\frac{1}{x})}{x^3}dx$,but I have never seen this notation before. It ...
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Cardinality of Primes in UFDs over Infinite Sets

As we know, there are infinitely-many Integer primes per, e.g , Euclid's proof. Is it also true that there are infinitely-many primes in every UFD defined over an infinite set? I tried to see if ...
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Does Infinity grows with respect to time? [closed]

Infinity is an idea that has no endpoint and it goes on forever. Does that mean Infinity grows with respect to time?
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Behavior of the sequences as n tends to infinity

I'm trying to figure out the behavior of some sequences in the following cases. The next two can be found in Hardy's "A Course of Pure Mathematics" (p. 131). If $\phi(n) \to +\infty$ and $\psi(n) \...
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How should I self-study set theory/cardinality?

So, I am an absolute beginner in mathematics; only being knolwdegable on some basic ideas in the subject. My interest in maths started only recently, while reading about set theory and cardinality (...
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Why is electric potential function in free space infinitely differentiable?

Electric potential function in free space of a continuous charge distribution $\rho'$ distributed over volume $V' \subset \mathbb{R}^3$ is denoted by: $\psi (x,y,z): \mathbb{R}^3 \setminus{V'} \to \...
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4answers
81 views

Evaluate $\lim_{n\to \infty} \sum_{k=0}^n \frac{\sqrt {kn}}{n}$

I'm not sure which would be the best way to compute this limit. As you might have observed, if you expand the infinite sum and rearrange some terms you get: $$\lim_{n\to \infty}\frac{\sqrt n}{n}\!\...
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1answer
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Finding $\lim\limits _{x\to\infty} \frac{1}{\frac{3}{x^3}-\frac{5}{x^2}+\frac{\sqrt[]{x}}{x^4}}$ [closed]

I'm having some tough time solving $$\lim\limits _{x\to\infty} \frac{1}{\frac{3}{x^3}-\frac{5}{x^2}+\frac{\sqrt[]{x}}{x^4}}$$
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1answer
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Limit of a sequence of infinities

I'm having trouble dealing with limits involving infinities. Suppose I have sequence $\{a_n\}_{n \in \mathbb{N}}$ and $a_n = \infty$ for all $n$. Then is the following true? $$\lim_{n \to \infty} a_n ...
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Whats the error in my continuum hypothesis “proof”

First of all, I just want to say that I know my "proof" is incorrect due to the continuum hypothesis being unprovable using the standard ZFC axioms. The reason I'm posting this is because I'm self ...
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2answers
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Does$ f(0) = 0, f'(x) > 0, f''(x) < 0$ imply that$ f'(0) $ is equal to infinity? [closed]

As the title describes it, I am interested in the following: Suppose you have a differentiable function $f:\mathbb{R}_+ \rightarrow \mathbb{R}_+ $, where $f'(x)>0, f''(x) <0$ and $f(0) = 0$. ...
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Is the cardinality of $R^3$ greater than the cardinality of $R^2$? [duplicate]

The origin of my question is : is it possible ( at least in the abstract) to represent a portion of (physical) space on a $2D$ map, say a square map? Such a representation would require (it seems to ...
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Why is infinity *infinity and infinity^infinity considered indeterminate?

I know that infinity is not a specific number,so we cannot apply normal algebric operations with it.But we can use the concepts of limits.So why is x^x or x*x (where x tends to infinity) indeterminate?...
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Limits to infinity of exponential functions

I am completly blocked trying to prove the solution of the limits below 1) $$ \lim _{m\to\infty}\left(\cos\left(\frac{x}{m}\right)\right)^m\\1 \quad \text{ for } a\to +\infty;\quad 0 \quad \text{ ...
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Evaluating $\sum_{n=1}^\infty\frac{1}{4n(2n+1)}$ [closed]

How to evaluate this sum, derived from "Lockdown math" by 3Blue1Brown? $$\sum_{n=1}^\infty\frac{1}{4n(2n+1)}$$
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1answer
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Why does this cross ratio equal infinity?

I'm currently studying linear fractional transformations and cross ratios and came across this in a book (this is translated from Korean, so I apologize if there are any errors or ambiguities): We ...
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3answers
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Why does $\sum \frac{1}{n^{1 + \epsilon}}$ converge?

The proof that the infinite sum of $\frac{1}{n}$ diverges seems to have a fair amount of breathing room. We group successive terms in quantities of increasing powers, starting with $\frac{1}{2}$, ...
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2answers
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How I prove that $\lim_{x\to\infty} x \sin x$ does not exist? [closed]

How I prove that this limit does not exist? $$\lim_{x\to\infty} x \sin x$$ I can't find two series that disprove this limit. What happens if I use a $2\pi n$ series? $\infty \times 0$ is indefinite. ...
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2answers
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Infinite sum power series

I would like to show $$ \sum_{r=0}^{\infty}\frac{1}{6^r} \binom{2r}{r}= \sqrt{3} $$ I have tried proving this using telescoping sum, limit of a sum, and some combinatorial properties but I couldn't ...
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Do any proofs/properties rely on the distinction between some uncountable size and a larger uncountable size, in order for the proof/property to hold?

(Sorry if my terminology is a bit imprecise; I'm trying to describe a rather vague notion in my head about when something "relies on" the distinction between finite/countable/uncountable, but I am ...
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1answer
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Evaluate $\int_{-\infty}^{\infty} e^{t \cdot x^{\alpha}} \cos(s \cdot x^{\alpha})dx$, expressing the answer in terms of $t$, $s$ and $\alpha$

One of the maths groups I'm apart of on Facebook posts (usually) daily maths challenges. Typically they act as small brain teaser for when I wake up and I can solve them without much trouble. However, ...
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1answer
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Are categories containing a countably infinite amount of sets small or large?

In category theory, small categories are those which all its objects and morphisms are sets and not proper classes. A category is small if it has a small set of objects and a small set of morphisms. ...
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Prove that if f is an entire function with lim(z tends to infinity)|f(z)| = infinity, then f has at least one zero.

I was trying to apply the limits involving point of infinity and putting z to 1/z. However, I could not proceed. Kindly help in my assignment.
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proving $\lim_{n\to\infty} |b_n| =1$

How do I prove that given $\lim\limits_{n\to \infty}\left(a_n\cdot b_n\right)=1$, that if $\lim\limits_{n\to \infty}\left(|a_n|\right)=1$, then $\lim\limits_{n\to \infty}\left(|b_n|\right)=1$? (I ...
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I think I may have found another large cardinal number.

The equation that I have come up with utilises the complex plane - something that I have not seen being used in many of the large cardinal numbers. By complex, as you probably know, I mean an equation ...
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1answer
26 views

sum of an infinite series, and extension of composite numbers to powers.

Consider all the positive numbers that can be expressed as a proper power of two integers (so that neither is 1). i.e. $2^2$, $2^3$, $3^2$, $2^4$, $5^2$... and so on. And let $c$ run over all of these ...
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1answer
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How do I prove that limit by definition?

How do I prove that $\lim\limits_{n\to \infty}\left(\frac{6n^2-8}{n^2-8}\right)=6$, by definition? I need to find $N_{\epsilon}$ such that for every $n>N_{\epsilon}$: $$\left|\left(\frac{6n^2-8}{n^...
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1answer
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How do people compartmentalize infinity when in the reals? [closed]

We see the common questions in math going through highschool. 1^infinity, undefined, why? Because infinity isn't defined in the reals? So we reorganize the question to include a limit, but we still ...
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Could divide by 0 be anything except plus or minus infinity

I know that division by zero is undefined, but let's put that aside for a moment. If you divide by a lower and lower amount, and approach it from a positive value, it becomes larger and larger. With ...
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1answer
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Is an uncountable infinity necessarily “bigger” than a countable infinity?

I'm not conversant in cardinality theory or set theory to formulate my question in much of a meaningful sense but I'll give it a try in hope of finding the right way to ask the question as an answer. ...
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Prove uncountability of R using an algorithm?

Let's say I want to prove uncountability of $\mathbb{R}$ using an algorithm (I will use Python). I will consider reals $0 \le x \lt 1$ and represent the decimal development of $x$ with a generator. ...
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1answer
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Average of $ \lim_{a \to -\infty} \int_a^\mu \frac{1}{\sigma\sqrt{2\pi}(\mu-a)} \exp \left(-\left(\frac{z-\mu}{\sqrt{2}\sigma}\right)^2\right)\,dz $

I'm trying to calculate the average of one half of the normal distribution curve. $$ \lim_{a \to -\infty} \int_a^\mu \frac{1}{\sigma\sqrt{2\pi}(\mu-a)} \exp \left(-\left(\frac{z-\mu}{\sqrt{2}\...
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4answers
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Finidng the infinity limit of $\coth$ function.

I have the function: $$ g(x) = \lim_{J \to +\infty} \frac{1}{2J} \coth(\frac{x}{2J}) $$ In the answers it gives: $$ g(x) = \frac{1}{2J}\frac{2J}{x} $$. I don't understand how the infinity limit ...
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Grandi's series manipulation allowing me to prove 1 = 0?

Let me start by saying I have a very basic background in math, so if I'm using the wrong terms, please feel free to point it out. I know straight away that this is wrong, I'm just curious to know ...
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Is $3^{\aleph_0}$ odd? [closed]

I'm actually interested in the range of answers. Is an odd number to the infinite power still odd? Must $3^{\aleph_0}$ be odd? I can't tell if the answer is yes or no. It seems like it should be ...
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1answer
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exp(a) goes to 0 and exp(a)exp(b) goes to infinity. What's a and b?

Suppose it is given that $e^{(a+b)} = e^a e^b \rightarrow \infty$ and that $e^a \rightarrow 0$. Is it possible to state mathematically how the values for $a$ and $b$ that fulfill this look like? ...
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Question Uncountable Set Binary Sequence Notation

In class, my professor mentioned that $\mathbb{N} \rightarrow \{0,1\}$ was uncountable, and I completely understand the argument. However, I am confused because others online gave a similar proof of $\...
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4answers
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how can I proof this summation diverge

$$\lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+.....+\frac{1}{n^2})$$ I tried to seperate it , and I could manage to proof this part does converge: $$\lim_{n\to\infty} (\frac{1}{n+1}+\...
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1answer
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A Question About the possible values of 1^infinity

if we have $$\lim_{x\to c}f(x) = 1 $$ $$\lim_{x\to c}g(x) = \infty $$ then can $\lim_{x\to c}f(x)^{g(x)} = -\infty $ ever be true? If so, what are some examples? If not, would it be different if $ ...
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Why does axiom of choice not imply the set of real numbers is countable?

The axiom of choice implies all sets can be well ordered. If that is true, you can well order the set of real numbers and the set of the integers. Now, why can one not just pair the set of real ...
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What is an weakly compact cardinal? [closed]

In set theory there are many types of cardinal numbers. One of them is the weakly compact cardinal. Wikipedia states the following about it: Formally, a cardinal $\kappa$ is defined to be weakly ...
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How to find the solution to a multi-inifite summation?

I'm familiar with techniques to solve a single infinite sum such as: $$ \sum_{i=1}^{\infty} (1 + i) \cdot \frac{1}{2^i} $$ Which ends up being equal to $3$, but I'm having trouble figuring out how ...
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2answers
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Limit of the form 0 times infinity

I'm trying to evaluate $\lim_{n\to\infty} n^3\ln\left(1+\frac{1}{n!}\right)$. It's $0\cdot\infty$ situation. I know that indeterminate forms can sometimes be evaluated using L'Hopital's rule. I ...
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Hilbert's Hotel Paradox: Guests moving to new room every day?

Suppose there are infinitely many coaches with infinitely many members in each coach. They stay at the hotel for infinitely many days. I know that guests can be accommodated using various methods like ...
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why shouldn't the two lengths be equal?

is it a paradox?? We know a function maps every POINT from set X to exactly one point in set Y . If a line is a collection of points then shouldn't the two lengths be same? Is this a paradox that ...
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finite set of countable sets is countable [duplicate]

I am trying to solve this using Hilbert's paradox. Say I have a collection of countable sets: $A_1,...,A_n$. I want to show that there is a bijective function from then union, $A=\cup{A_i}, 1\leq i\...
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1answer
34 views

$\lim_{n\to 1}(\frac{1}{1-n})=\infty$

$\lim_{n\to 1}(\frac{1}{1-n})=\infty$ When $n=1$, the equation, of course, becomes undefined, as it becomes $\frac10$ I know this can be proven since $\frac{1}{1-n}=\sum_{k=0}^{\infty}n^k$, which ...
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2answers
81 views

How can an event with probability $0$ be possible?

Consider a dart board that is represented by a unit circle centred at the origin. Each dart lands at a singular point within the circle (or on its outer edges). Arguments that the probability of the ...
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1answer
31 views

how to prove that a set is not bounded above?

I got stuck in proving that $A=\{yn:n\in\mathbb N,y\in(1,\infty)\}$ is not bounded above? (without of course using lim)?
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how to prove that a set is not dense?

Let $A$ be a set of real numbers such that 𝐴⊆(1,∞) and dense in $(1,\infty)$, I would like to prove that $B= \left\{\frac{a}{(a+1)n^2}|a∈A,n∈N\right\}$ is not dense in $[1,0]$. Well I'm having some ...

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