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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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
105 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
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
73 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
330 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
184 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
24 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
192 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
77 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
vote
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 = ...
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
61 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
102 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
74 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
118 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
72 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
64 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
78 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
78 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
115 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
535 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
103 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
256 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
338 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
492 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. ...
0
votes
2answers
76 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
103 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
82 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
76 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
68 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 ...
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
39 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 ? ...
7
votes
6answers
299 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
174 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 ...
1
vote
1answer
787 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
118 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
52 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
133 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
139 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 ...