Somewhere beyond the numbers lies the concept of Infinity. But what exactly does "infinity" mean? What rules does it obey? What interesting properties does it have?

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Are there fewer reals on $(0, 1)$ than on $(1,\infty)$?

I know that the cardinality of the sets of real numbers $(0, 1)$ and $(1, \infty)$ are equal. So what is the fallacy in this argument? For every real on $(0, 1)$, we can add any integer $n$ to it ...
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1answer
29 views

If $\dim(V_F)$ is Infinite, Does It Follow $\dim(\operatorname{Hom}(V_F, W_F)) \ge |F|$?

Part of the proof that $\dim(V^*_F) > \dim(V_F)$ for an infinite dimensional space is that $\dim(\operatorname{Hom}(V_F, F)) \ge |F|$ (i.e $\dim(V^*_F) \ge |F|$). See for example Dual space ...
2
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6answers
105 views

More numbers between $[0,1]$ or $[1,\infty)$?

There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are ...
3
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1answer
55 views

Improper integral: why $\int_0^1(x^2+ x^{1/3})^{-1}\,dx$ is convergent and not $\int \frac{1}{x^2}\,dx$ ???

How do I show that $\int_0^1(x^2+ x^{1/3})^{-1}\,dx$ converges? I assume you show it on $(0,1]$. Can't seem to get my head around why this would be true.
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0answers
37 views

For Vector Spaces V and W with one Infinite Dimensional , is Hom(V, W) Isomorphic to Hom(W, V)?

If V and W are both finite then clearly Dim (Hom(V, W)) = Dim(V).Dim(W) = Dim(Hom(W, V)) so they are isomorphic. I'm not so sure if one is infinite. An "infinite matrix" construction for a linear ...
3
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2answers
69 views

What are some good reasonably rigorous texts on the mathematics of infinity?

The Infinite Book is too light and not focused enough on the mathematics of infinity, and Everything and More: A Brief History of Infinity has too much focus on the history of infinity instead of the ...
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2answers
43 views

What is the correct mathematical notation for something comprised of the sum of constituents n where n is infinite?

I am trying to figure out what the correct mathematical notation would be for something like the following: I want to describe that the value V of a company is equal to sum of parameters P at any ...
3
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2answers
69 views

Is my argument correct to solve this textbook problem?

The problem is from M.Bona's "A Walk through Combinatorics", Ch1 Prob 13: There are infinitely many pieces of paper in a basket, and there is a positive integer written on each of them. We know ...
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3answers
61 views

Find a limit without using L'Hopitals rule 9

Can someone please show me how to do this without using L'Hopitals rule: $$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x$$ I know the limit is $e^a$, but I would like to know the steps taken to ...
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1answer
39 views

What is the limit of the below functions when n tends to inifinity?

What is the value for the functions in the image when limit n tends to infinity?. Also what is the asymptotic complexity (big $O$ notation) for all the four functions?. $$\begin{aligned}f_1(n) &= ...
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2answers
112 views

derivative of x^x^x… to infinity?

I am a 12th grade student, and I am afraid that in realistic terms this question might not even make sense because of the infinities that have to be dealt with. However, in my attempt to calculate ...
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3answers
72 views

Why does $e^{\frac10}\neq e^{\frac1{-0}}$?

I was unable to explain why this fails? I asked to it many peers and they too can't. I faced this situation when solving a kind of integration problem. Consider $x=-x$ Then $x=0$ That is, $0=-0$ ...
0
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3answers
226 views

Do different infinities have different base representations?

I've been wondering about different types of infinity; e.g., $\aleph_0,\aleph_1$, e.t.c.; where $\aleph_0$ represents the smallest infinity, the countable infinity (e.g., the cardinality of the ...
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3answers
84 views

Proof of $\sum_{x = 1}^\infty \frac{1}{x}$'s divergence by absurdity?

(From this site.) The following argument purports to show that the series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \dots = 0$. It begins with the harmonic series. $$ \begin{aligned} \sum ...
2
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1answer
148 views

determinant of infinitely large matrix by decomposition

Read the too long didnt read version in bold before going into the finer detail. The overall point is that when I decompose this matrix to try and find its determinant I get an answer that doesn't ...
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0answers
77 views

0's reciprocal (Theoretical) [closed]

Background: I am in 8th grade and I like to study around advanced mathematical subjects. However, I do not know enough to be sure in my conjectures. Therefore, I would like your help. I have a ...
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2answers
56 views

Difference between intersection of infinite sets having finite, and having infinite elements

I could find individual answers for both of these, but can someone compare how being a finite set or an infinite changes the final outcome? (a) If A1 ⊇ A2 ⊇ A3 ⊇ A4 · · · are all sets containing an ...
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2answers
182 views

Rolling two dice, what is the probability that two consecutive $7$s happens earlier than a $12$?

Alice and Bob are playing a game involving two dice. If a sum of 12 appears, Alice wins and they stop playing. If a 7 appears twice in a row, Bob wins and they stop playing. What is the probability ...
2
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5answers
261 views

Is there an object in reality that is proven to be uncountable? [closed]

I've always wanted to come up with a fairly concrete example of an object or realistic set that could be uncountable. Most of the sets I can think about, even the hugest ones, are always countable. ...
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3answers
272 views

Why does Wolfram Alpha say that $n/0$ is complex infinity?

I typed a number divided by 0 on Wolfram Alpha and thought that it would say "undefined". However, when I pressed enter it told me that the answer is complex infinity. I have always been taught ...
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3answers
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Trying to show that $\sum_{1}^{\infty} \frac{n^n}{n!} $ diverges.

I have been trying to prove this using the ratio test $|\frac{A_{n+1}}{A_n}|$ , which leads me to this expression: $$\left|\frac{(n+1)^{n+1}}{(n+1)!}\cdot ...
3
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1answer
68 views

Evaluate the limit $\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $

It seems reasonable to assume that $$\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $$ goes to zero but I can't figure out how to prove it.
7
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1answer
49 views

Infinite solutions for $(\frac{n+1}{n})^a\cdot (\frac{m+1}{m})^b = 2$

Given $(\frac{n+1}{n})^a\cdot (\frac{m+1}{m})^b = 2$ where a, b, n, and m are all positive integers, are there infinitely many solutions $(a,b,n,m)$?
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5answers
114 views

Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ [duplicate]

I'm completely stuck evaluating $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ how would I go about solving this?
3
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1answer
48 views

Existence of infinite set and axiom schema of replacement imply axiom of infinity

I'm self-teaching an intro to set theory course, and came across this exercise: Show that the existence of an infinite set is equivalent to the existence of an inductive set. For the notion of ...
2
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5answers
94 views

To evaluate $\lim_{n\to\infty} \dfrac{10^n}{n!} .$

I am having a lot of trouble evaluating $$\lim_{n\to\infty} \dfrac{10^n}{n!} .$$ I know intuitively that $n!$ is much larger and this expression should go to zero but I just can't figure out how to ...
2
votes
2answers
42 views

recursive sequences bounded above and their limits at infinity

Define a sequence $\langle a(n)\rangle$ recursively by $a(1)=\sqrt{2}$ and $a(n+1)=\sqrt{2+a(n)}$ $(n>0)$. a)by induction or otherwise show that the sequence is increasing and bounded above 3. ...
0
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1answer
36 views

bounded sequences and limits at infinity

Show that if ${a_n}$ and ${b_n}$ are sequences for which $\displaystyle\lim_{n\to\infty}a_n=0$ and ${b_n}$ is bounded, then $\displaystyle\lim_{n\to\infty}a_nb_n=0$ Sorry I'm on my phone and I'm not ...
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2answers
76 views

Probability over an infinite set. Of two numbers.

Suppose there is a man, who chooses two completely random numbers. So they can be equal too. And they can be only positive real numbers. One of them can even be $285294.38285967281$ or anything ...
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2answers
58 views

infinite series and proof of sum using induction.

Consider the series: $$ \sum_{i=1}^\infty \frac{i}{(i+1)!} $$ Make a guess for the value of the $n$-th partial sum and use induction to prove that your guess is correct. I understand the ...
4
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2answers
61 views

Infinity - infinity calculus

$$\lim_{x\to\infty} (x-1)e^{-1/x}-x$$ I know that this limit equals $-2$ but I don't know how to prove it. I can only get to $\infty-\infty=?$
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2answers
41 views

Are there any certainties of properties of infinity?

I have been considering the properties of infinity and applications to various areas of maths and was hoping to get some opinions of more seasoned mathematicians than myself. One geometric ...
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1answer
44 views

Limit of functions involving trigonometry as n approaches infinity

By graphing these functions, I know that P(n) approaches pi as n tends towards infinity. However, is there a mathematical way for proving this? I am doing a maths exploration on Archimedes' ...
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2answers
49 views

limit as $x$ approaches infinity of $\frac{1}{x}$

How can I show $\lim_{x \to \infty} \frac{1}{x} = 0$ using epsilon delta proof. Its pretty obvious that the limit is zero, but I am still new at epsilon proofs.
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2answers
37 views

Find $\lim_{x\to \infty} \sin\left(\frac 1x\right) $

Show that $$\lim_{x\to \infty} \sin\left(\frac 1x\right) = 0$$ I don't even really know where to start on this question. I know that the limit definition at infinity is: for all $\epsilon>0$, ...
3
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0answers
64 views

How does $\ln(x)$ blow up at $0$ and $\infty$.

In general: How do I figure out how fast a function blows up at a certain point or infinity? How fast does $\ln x$ blow up at $0$? Does it blow up as fast as $1/x$, $1/x^2$, or maybe faster than any ...
4
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1answer
94 views

Is the set of real numbers really uncountably infinite?

The proof that the set of real numbers is uncountably infinite is often concluded with a contradiction. In the following argument I use a similar proof by contradiction to show that the set of ...
5
votes
1answer
62 views

Perfectly Round Sphere on Perfectly Flat Floor

I know that in practice it may be practically impossible to create the following situation, but suppose I did place a perfectly round ball of radius r (not that I think radius r is relevant) on a ...
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2answers
48 views

Evaluate the following improper integral.

$$ \int^{+\infty}_{-\infty} \frac{x\sin 4x}{x^2-4x+8}dx \, $$ My Thoughts: I know that I should start by changing the integral to: $$ \int^{+\infty}_{-\infty} ...
0
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1answer
47 views

Another flavour of Hilbert's Hotel

Many of you have probably heard about Hilbert's Hotel problem. Mr Hilbert owns a hotel with countably infinite amount of one-bed rooms. All the rooms are, of course, taken. A (finite or infinite) ...
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0answers
41 views

Laplace transform and “imaginary infinity”

I was recently studying Laplace transform for the first time, and I'd like to ask the following thing: there was an integral with limit of integration, something like that: a+j×infinity, j the ...
2
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3answers
45 views

Construct a non-linear function that shows that the intervals $[2,4]$ and $[10,22]$ have the same cardinality

Using something other than a linear function, show the intervals $[2,4]$ and $[10,22]$ have the same cardinality. I don't quite know where to start with this problem, or what key factor is necessary ...
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4answers
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Need assistance solving a limit question without applying l'hopital's rule

This is the question: $$\lim_{x \rightarrow-\infty} \frac{|2x+5|}{2x+5}$$ I know the answer is $-1$, but can someone go through the steps and explaining it to me?
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0answers
38 views

find the supremum

Hi I'm trying to figure out for which values of $w$ of $u(x,t)$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is ...
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1answer
38 views

If $A$, $B$, $C$ are any infinite sets then is $|A|=|B|$ and $|A|=|C|$ $\Longleftrightarrow |A|=|B\cup{}C|$?

Suppose we have three sets $A$, $B$, and $C$ that we know are infinite sets, but we do not know anything else about the cardinality of $A$, $B$, and $C$. Is $|A|=|B|$ and $|A|=|C|$ ...
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3answers
72 views

Uncountable infinity

The "number" of real numbers in $[0,1]$ is uncountably infinite, just as the "number" of real numbers in $[0,10]$ is uncountably infinite. However, my intuition would tell me the second interval has ...
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0answers
21 views

Compute a sum with finite and infinite elements

I would like to compute the following summation: $$ s = \sum_{i=1}^n a_i \, \Phi^{-1}(u_i) $$ where $\Phi^{-1}$ is the inverse of the standard Gaussian distribution function, $a_i$ are some real ...
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1answer
83 views

Question about infinity

That might be a silly question, but here goes: I see a lot of "big numbers" in physics, such as the size of the state space of all the particles in the visible Universe, and those numbers can be ...
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2answers
36 views

Infinite averages

If you wanted to find the average of infinite items, what would it be? Would an estimate of the first x averaged be a good estimate? Or would the value be nearly zero because you are dividing by ...
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0answers
41 views

How does one prove that two sequences are equal at infinity?

I came up with $e = \sum_{n=0}^\infty \frac 1 {n!}$ (see here) I am now trying to prove that this is equivalent to $\lim_{n\to \infty} {(1+\frac1 n)}^n$ In general, how would one go about such a ...