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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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 ...
0
votes
3answers
107 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. ...
7
votes
7answers
519 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
63 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
238 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. ...
-9
votes
2answers
212 views

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

Hi everyone :) - this question has been closed for quite some time. I have no way of deleting it because it has answers so the site won't let me, but please stop down-voting haha. I've learned my ...
1
vote
3answers
368 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
48 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
83 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. ...
0
votes
2answers
74 views

Does $\frac{1}{x}=0$ have a solution?

Does $\frac{1}{x}=0$ have a solution ? since any number multiplied by 0 equals 0,this equation has no solution in elementary math. I wonder is there a solution in higher mathematics.
0
votes
0answers
22 views

Probability of getting black or white

If I have a bag of black and white marbles, say 10 black and 10 white, I have a 50% chance of either getting black or white. What if the bag were infinite? Can one talk about probability when ...
2
votes
2answers
98 views

Trying to find $\lim_{x\to 0^+} \frac{x^2\sin(1/x)}{\sin x}$, I get $\frac{\infty}{0}$, what is that?

I'm trying to find $\lim\limits_{x\to 0^+} \dfrac{x^2\sin(1/x)}{\sin x}$, I get $\dfrac{\infty}{0}$. If $ \frac{\infty}{0} $is not an indeterminate form (like $ \infty \times 0, 1^\infty, ...
-2
votes
1answer
79 views

Weird problem with L'Hopital's rule: $\lim\limits_{x\to 0^-} x^3e^{1/x}$

$$\lim\limits_{x\to 0^-} x^3e^{1/x}$$ If I plug in zero, I get $0\cdot\infty$. So, this, I thought was a hint to try to rewrite the problem to try to get $\frac{\infty}{\infty}$ or $\frac{0}{0} so ...
0
votes
1answer
73 views

“Not” indeterminate form problems

"...are not indeterminate forms. Find the following by inspection:" $\displaystyle\lim_{ x\to \pi/2} (\cos x)^{\tan x}$ and $\displaystyle\lim _{x\to \pi/2} [ (2/\pi-2x) + \tan x ]$ These are ...
1
vote
2answers
59 views

Solve L'Hopitals problem

$$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x}{{\sec^2 3x}} $$ I used LH: $$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x \tan x}{6\sec 3x \sec 3x \tan 3x}$$ then: $$\lim_{x\rightarrow ...
1
vote
1answer
47 views

Use L'Hopital's with this problem?

The problem is: $$\lim_{x\rightarrow 0^+} \left(\frac{1}{x}\right)^{\sin x}$$ I know the answer is $1$ because I checked with my graphing calculator, but how exactly do I get there? I got this far: ...
2
votes
2answers
66 views

$\lim_{x \to \infty} \frac{\sqrt{x^2 -1}}{2x+1}$

So the question is: $$\lim_{x \to \infty} \frac{\sqrt{x^2 -1}}{2x+1}$$ First of all, I know we have to use Lhopital's rule. However, I just don't know how. Second of all, I thought in the end we ...
1
vote
0answers
42 views

The sum of the infinite series [duplicate]

The sum of the infinite series: $$ \frac{1}{2} +\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +\cdots$$ I am able to find ...
0
votes
2answers
54 views

I need advice on calculating this limit of a function resulting in $-\frac14\pi$

I am looking for advice on solving this limit of a function. I am struggling to find the correct process: $$\lim\limits_{x\to-\infty}\operatorname{arccotg}\frac{x}{(x^2-4)^{\frac12}}$$
0
votes
2answers
37 views

What is the meaning of Right Hand Limit at $\infty$?

For a limit to exist, the left hand limit must equal the right hand limit. That is, $$\lim_{x\to c^+}f(x)=\lim_{x\to c^-} f(x)$$ However, if $x\to\infty$, then what does the right hand limit mean ? ...
6
votes
5answers
141 views

dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

I have a question about this integral $ \displaystyle \int_{-\infty}^{+\infty} \delta'(x-3)e^{x^2}dx $ by integration by parts I get; $ \displaystyle ...
0
votes
4answers
110 views

Why Zero divided by Zero is undefined and not Infinity [duplicate]

apologize in advance if this is a duplicate, but I found a lot questions related to this but none answering this specific question. My logic is: let's consider division the opposite of ...
0
votes
0answers
41 views

Good papers on Cantor's infinities?

I'm trying to get a mathematical treatment on Cantor's infinities, but can't seem to find papers merely by googling. Any recommendations?
1
vote
1answer
557 views

An infinite dictionary: countably infinite or uncountably infinite?

This question concerns Ian Stewart's "Hyperwebster", an uncountable dictionary. Say a publishing company wants to publish every possible permutation (of any length) of the characters A-Z. The ...
0
votes
1answer
55 views

Comparing Infinities

Is it ever possible to say that $\infty = \infty$? For example, does the number of odd numbers ($\infty$) equal the number of even numbers ($\infty$)? Does does the number of odd numbers ($\infty$) ...
1
vote
2answers
45 views

Proving existence of numbers with intermediate value theorem

How do you use the intermediate value theorem to prove the existence of numbers? For example, with $f(x) = c^2 = 2$, how can I prove that $\sqrt2$ or a positive number $"c"$ such that $f(x)$ is true ...
2
votes
1answer
22 views

The Limit of an Integral Containing Exponentials

I am unsure how to show this. Suppose $\delta(s)$ defined on $(-\infty , s_*)$ is increasing and satisfies $\lim _{s\rightarrow s_*} \delta = \lim _{s\rightarrow s_*} \frac{d \delta}{d s} = \infty$ ...
0
votes
1answer
108 views

Coupon collector problem doubts

The Coupon Collector problem off Wikipedia: Suppose that there is an urn of $n$ different coupons, from which coupons are being collected, equally likely, with replacement. How many coupons do you ...
-3
votes
2answers
134 views

Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? [closed]

Here is a small intuition why it should be the later. Let $\omega$ be the number of all natural numbers. Then what is the smallest real number? We can write reals in binary form. Usual logic would ...
29
votes
8answers
2k views

Can a sequence have infinitely many limits among its subsequences?

Suppose we have an extended real (countably) infinite sequence $(x_n)$. Then consider all of its possible subsequences $(x_{n_k})$. We could then consider the set $$A = \{a\in ...
0
votes
1answer
32 views

Is there a relationship between the unbounded infinities and uncountable infinities? [duplicate]

When a function f increases without bound we say $f(x)=\infty$. How does this idea relate to, if at all with the infinite sets we study in set theory? To give a better understanding of why I'm ...
3
votes
2answers
97 views

Double Summation Trick

I have seen a couple of times the trick where $\displaystyle\sum_{i=1}^\infty \sum_{j=i}^\infty f(j)$ becomes $\displaystyle\sum_{j=1}^\infty \sum_{i=1}^j f(j)$ How does this work? I am so confused. ...
2
votes
1answer
37 views

Logic question requiring axiom of choice

Predicting Real Numbers Regarding the above question, the solutions require creating classes of sequences with representative sequences. How are those sequences constructed? How is it possible to ...
6
votes
2answers
740 views

Why do we distinguish between infinite cardinalities but not between infinite values?

More specifically, why are we "allowed" to denote $|\mathbb{N}|<|\mathbb{R}|$ but not $\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$? Can we distinguish between "countable ...
3
votes
2answers
54 views

Alternate Axiom of Infinity

The Axiom of Infinity states that there is a set $S$ containing $\varnothing$ such that if $x$ is an element of $S$ then so is $x\cup\{x\}$. Is the following variant equivalent? There exists a ...
1
vote
3answers
247 views

Two similar method to calculate one equation get different answer

Method1:$$\lim_{x\rightarrow0}({\frac{e^x+xe^x}{e^x-1}}-\frac1x)=\lim_{x\rightarrow0}({\frac{e^x+xe^x}{x}}-\frac1x)=\lim_{x\rightarrow0}(\frac{e^x+xe^x-1}{x})=\lim_{x\rightarrow0}(2e^x+xe^x)=2$$ ...
1
vote
0answers
82 views

Is there a change-of-variables solution for integrals from negative infinity to a constant?

I found a fantastic and generalizable substitution technique for computing definite integrals that go to infinity from either negative infinity or a constant, regardless of the function (sorry for the ...
2
votes
2answers
186 views

Is $\frac00=\infty$? And what is $\frac10$? Are they same? Does it hold true for any constant $a$ in $\frac{a}0$ [duplicate]

I know that $\lim_{x\to0}\frac{x}{x}=$ 1. But in my text book, it is written that it is $\infty$ and even $\frac10=\infty$. But how is it possible? And are they both same? What is the difference ...
1
vote
2answers
53 views

What is the value of $0\times \infty$? (in $[0, +\infty]$)

In $[0,+\infty)$, $0^+\times +\infty$ can be any number in $(0,+\infty)$ so is undetermined; (in which $0^+$ means when a variable approaches to $0$). Because $\lim_{x\rightarrow ...
5
votes
4answers
186 views

Can a number have an uncountably infinite amount of digits?

I'm an extreme mathematical layman, so please excuse the probable ignorance and awkward phrasing of this question. Is there such thing as a kind of number which has an uncountably infinite amount of ...
5
votes
7answers
2k views

One divided by infinity is not zero? [duplicate]

I know that $\frac{1}{\infty}$ is undefined. But my question is - can we say that $\frac{1}{\infty}\neq0$ ? I've got some idea how to explain that: Let's say we have a random-number generator that ...
1
vote
2answers
115 views

Third axiom of Kolmogorov axioms

Let us define for a countably infinite set $S$ of real numbers that can be enumerated as $x_1,x_2,\cdots$, $$P(S) = \sum_{x \in S}p(x) = \sum_{i=1}^\infty p(x_i) = \lim_{n \to \infty}\sum_{i=1}^n ...
1
vote
2answers
85 views

How to find the limit of a sequence?

Question: If $0 < x < \frac{\pi}{2}$ and $f_k(x) = \tan(x)+\frac{1}{2}\tan(x/2)+ ...+\frac{1}{2^k}\tan(x/2^k)$. In Sigma Notation: $$f_k(x) = \sum_{n=0}^k \frac{1}{2^n}\tan\frac{x}{2^n}$$ ...
1
vote
2answers
96 views

$\ln{\left(\frac{1}{0}\right)} = -\infty$?

I have shown it using a theorem that I made, but I am not sure, as $\lim_{\alpha \to 0^{-}}{\left(\frac{1}{\alpha}\right)} = -\infty$, and $\lim_{\alpha \to 0^{+}}{\left(\frac{1}{\alpha}\right)} = ...
24
votes
2answers
927 views

W. Mückenheim claims a severe inconsistency of transfinite set theory; true? [closed]

The abstract for a paper on arxiv.org (http://arxiv.org/pdf/math/0408089v3.pdf) reads (with my emphasis): "Transfinite set theory including the axiom of choice supplies the following basic theorems: ...
-1
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3answers
321 views

Is infinity an element of the natural numbers? [duplicate]

I am wondering, whether $\infty \in \mathbb{N}$.
18
votes
4answers
1k views

Are there more transcendental numbers or irrational numbers that are not transcendental?

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it ...
2
votes
1answer
37 views

Group action of $GL(2, F)$ on the projective line $P(F)$

I refer to section 8.3, page 119 of Algebra, A Computational Introduction by John Scherk. It is about group action of $GL(2, F)$ on the projective line $P(F) = F \cup \{\infty\}$. Given a matrix ...