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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35
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10answers
43k views

Why is Infinity multiplied by Zero not an easy Zero answer?

I did a bit of math at school and it seems like an easy one - what am I missing? $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$ $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ ...
7
votes
1answer
45 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 $(a,b,n,m)$?
6
votes
2answers
132 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
votes
1answer
25 views
17
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5answers
2k views

How many different sizes of infinity are there?

It's pretty straightforward to say that there is an infinite number of different sizes of infinity, but then I thought, "What size of infinity is that?" My thoughts are that the number of unique ...
1
vote
3answers
82 views

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 ...
1
vote
2answers
31 views

Convergence of this alternating series: $\sum_{k=0}^\infty \frac{(-1)^k}{(k+1)C^k} = C \log \frac{C+1}{C}$

I "heard" the following formula for any $C \ge 1$: $\sum\limits_{k=0}^\infty \dfrac{(-1)^k}{(k+1)C^k} = C \log \dfrac{C+1}{C}$ Is it correct? What would be a proof?
2
votes
5answers
236 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. ...
2
votes
1answer
403 views

What is infinity in complex plane and what are operation with infinity extended to complex numbers?

For a real number $a$, $$\infty + a = \infty,$$ and if $a$ is positive, $$\infty \cdot a = \infty$$ What is $\infty + a$ and $\infty*a$ if $a$ is non-zero complex number, where $\infty$ is real ...
3
votes
1answer
76 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 ...
1
vote
3answers
190 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 ...
4
votes
1answer
409 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
5
votes
1answer
47 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 ...
1
vote
5answers
104 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?
2
votes
1answer
59 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.
2
votes
5answers
92 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
0answers
27 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 ...
1
vote
4answers
63 views

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?
1
vote
2answers
31 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
votes
1answer
32 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 ...
2
votes
2answers
49 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 ...
0
votes
2answers
47 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
votes
2answers
53 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 $\infty-\infty=?$
1
vote
2answers
114 views

$\lim_{n\to \infty} {\sqrt[n]{n^{-n}+2^n}}$?

$\lim_{n\to \infty} {\sqrt[n]{n^{-n}+2^n}}$ Intuitively, this seems like it should equal 2, but how would one go about showing this? I have tried factoring this somehow, but whatever form I get it in ...
0
votes
2answers
25 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 ...
0
votes
3answers
104 views

Discount in Infinity

I fully understand that $0.9999...$ mathematically will equal 1.0 exactly as the repeating decimal continues infinitely. This would be contrary to conventional logic where such a fraction is ...
1
vote
1answer
36 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' ...
0
votes
1answer
1k views

why does e raised to the power of negative infinity equal 0?

Why is it that e raised to the power of negative infinity would equal 0 instead of negative infinity? I am working on problems with regards to limits of integration, specifically improper integrals ...
0
votes
1answer
38 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) ...
1
vote
2answers
44 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.
1
vote
2answers
36 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
votes
0answers
54 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 ...
0
votes
0answers
61 views

Why is $y + 1$ infinite?

This is related to SO question : http://stackoverflow.com/questions/30150877/why-does-this-cause-ghci-to-hang but I'm having difficulty understanding why Haskell enters an infinite loop but since ...
1
vote
2answers
44 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} ...
34
votes
8answers
2k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
3
votes
3answers
432 views

Comparing infinite numbers

Suppose you have 2 infinite numbers, say $A$ and $B$. $A$ is an element of the hyperreals, so that $A$ is greater than every real number. $B$ is the size of the set of natural numbers, $\aleph_0$ ...
141
votes
12answers
51k views

What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
0
votes
1answer
44 views

Is the number of points on a plane larger than the number of points on a line?

The number of points on a line is uncountably infinite. The number of lines on a plane is uncountably infinite. It seems like it follows that there would be an uncountably infinite number of points on ...
0
votes
0answers
28 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 ...
-6
votes
6answers
160 views

All numbers are undefined? [closed]

Notice the following set of equations $$\infty = \infty -x$$ $$x = \infty - \infty$$ $$x = \text{Not defined}$$ here $x$ can be any real number and hence all real numbers are undefined. This is ...
2
votes
3answers
38 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 ...
19
votes
10answers
7k views

Does infinity and zero really exist?

I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please ...
0
votes
3answers
304 views

Why I can't calculate $0*log(0)$ but can $log(0^0)$

I got this doubt after some difficult in programming. In a part of code, i had to calculate: $$ x = 0 * Log(0) \\ x = 0*-Inf $$ and got $x = NaN$ (in R and Matlab). So I changed my computations to ...
0
votes
0answers
33 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 ...
0
votes
1answer
36 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 $|A|=|C|$ ...
1
vote
3answers
57 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 ...
0
votes
0answers
19 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 ...
1
vote
1answer
79 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 ...
1
vote
2answers
29 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 ...
1
vote
0answers
35 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 ...