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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6
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3answers
282 views

$\frac{1}{\infty}$ - is this equal $0$? [duplicate]

I've seen that wolfram alpha says: $$\frac{1}{\infty} = 0$$ Well, I'm sure that: $$\lim_{x\to \infty}\frac{1}{x} = 0$$ But does $\frac{1}{\infty}$ only makes sense when we calculate it's limit? ...
0
votes
1answer
165 views

The proof of the infinity base of $\mathbb{R}^{\infty}$

We know that a finite basis of the finite-dimensional space $\mathbb{R}^n$ is $$ \{(1, 0, 0, 0,\ldots,0),\:(0, 1, 0, 0, 0,\ldots,0),\:(0, 0, 1, 0, 0, 0,\ldots, 0),\:\ldots,\:(0, 0, \ldots, 0, 0, 0, ...
3
votes
4answers
223 views

Hilbert's Hotel and Infinities for Pre-university Students

Hilbert's paradox of the grand hotel is a fun and exciting ground to base a talk on the set theoretic concept of infinity for interested students - even in middle- and high school. However, it does ...
0
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1answer
105 views

Proving the product of two series diverges to infinity.

Proving the product of two series diverges to infinity, given that one series (An) converges to a limit L and (Bn) diverges to infinity, I have to prove that the product of the two series (AnBn) ...
0
votes
3answers
146 views

Why does $0,\bar{9}$ equal $1$? [duplicate]

I am finding hard to understand why $0,99999..... = 1$ I have the following proof: Let $x$ be $0,9999...$ then $10x = 9,999...$ So $10x - x = 9,999 - 0,9999$ $9x = 9 \rightarrow x = 1$ From a ...
2
votes
2answers
161 views

Is an infinitely small percentage of infinity infinite?

I'm not a mathematician, but this question intrigues me: Is an infinitely small percentage or part of infinity infinite? Do the two infinities "cancel out", leaving you with a real number? It seems ...
0
votes
1answer
51 views

About the order of infinity of $\Re^n$

I would like to ask a question about the order of infinity of the $n$-dimensional space $\mathbb{R}^n$. I am not sure whether I use the appropriate notation/mathematical language or not - please ...
4
votes
1answer
164 views

Can we determine $A= 1!+2!+3!+…$'s digits starting from last?

After reading a bit about p-adic numbers, I came up with an idea. We know that for every natural number $k$, there exists a natural number $n$ so that for every $m>n$, there are at least $k$ zero ...
1
vote
1answer
49 views

Show A is countable infinity

One more question about set theory: $A\subseteq R$ is an infinite set of positive numbers. Assume there is a value $k \in Z$ such that for any $B \subseteq A$: $\sum_{i=0}^\infty b(i) \le k$ where ...
3
votes
3answers
102 views

Infinite compact subset of $\mathbb{Q}$

Can I find an infinite set in $(\mathbb{Q},\mathcal{T}_e|_\mathbb{Q})$ which is compact?
0
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2answers
123 views

Does infinity have a limit?? [closed]

Infinity being the far extent that the numerical system can stretch,can we say that infinity is actually a limit or infiity has another limit?
-2
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3answers
50 views

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ $(A_n+B_n) \Longrightarrow \infty$

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ $(A_n+B_n) \longrightarrow \infty$ How do you prove it?
2
votes
2answers
103 views

Limit Involving Factorials

How would you go about calculating $$ \lim_{x \to \infty} \frac{x!}{(x - k)!} $$ for some constant $k > 0$?
1
vote
0answers
55 views

Prove infinity arithmatics

How do you prove $ \infty * (-\infty) = -\infty$ or $ \infty +\infty = \infty$? I thought it is an axiom, but have been there's is proof for that.
0
votes
2answers
108 views

How does $1^\infty=\infty$?

I remember hearing in school long ago that $1^\infty=\infty$. I was just wondering if anyone could explain this in laymen's terms?
86
votes
17answers
12k views

Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
0
votes
1answer
59 views

Is the set of all sums-of-rationals-that-give-one countable?

Some (but not all) sums of rational numbers gives us 1 as a result. For instance: $$\frac12 + \frac12 = 1$$ $$\frac13 + \frac23 = 1$$ $$\frac37 + \frac{3}{14} + \frac{5}{14} = 1$$ Is the set of all ...
0
votes
1answer
49 views

Prove $\mathop {\lim }\limits_{x \to \pm \infty } {a \over x} = 0$ [closed]

How do you prove: $\mathop {\lim }\limits_{x \to \pm \infty } {a \over x} = 0$
2
votes
2answers
265 views

Does The Monty Hall Problem Still Apply With Infinite Doors?

Here's been a bunch of questions on the Monty Hall problem, so I'll assume people know the basics. This answer helped clarify a few things for me, but talking with some colleagues yesterday, someone ...
1
vote
0answers
55 views

A weird infinity problem. [duplicate]

A weird infinity problem. I saw this on youtube but could not understand it: Let us add 1 + 2 + 4 + 8 + 16 + ... up to infinity x=(1+2+4+8+...) = 1(1+2+4+8+...) = (2-1)(1+2+4+8+...) = ...
0
votes
1answer
58 views

If $a < b$ and $b = \infty$, then $a < \infty$?

A very simple question, but I am not sure for this moment. I have a strict inequality $a < b$. And I prove that $b = \infty$, say $b$ is an integral. Does this prove that $a < \infty$, that is, ...
2
votes
3answers
71 views

negative and positive infinity

This is a weird question that I thought of and I was wondering if I could get some help. So normally $\frac{1}{x} = \frac{1}{y}$ then x and y would have to both be the same number, but with infinity ...
4
votes
2answers
133 views

How do mathematics define a point?

I have a serious doubt. How do mathematicians define a 'point' in a space or a plot? If we have a clear explanation for a 'point' , I think my doubt on infinitesimals and infinity will be clarified.
1
vote
1answer
57 views

3-D function that follows an inverse square law, but has an overall integral equal to a constant

I'm currently trying to figure out a 3D function which follows the "inverse square law" along any given ray drawn from 0,0,0 coordinates, but whose -inf..inf integral over all arguments converges. ...
0
votes
1answer
63 views

Delta function that obeys inverse square law outside its (-1; 1) range and has no 1/0 infinity

Does anybody know if such function exists? As I understand it, the function $$\frac{1}{x^2}$$ itself could be used as a delta function if it had no 1/0 infinity. That is why I'm in a search of an ...
1
vote
4answers
97 views

Counting down by halving to 0

Say that you are counting down from 10. You say how long is left after half the amount of time you said how long was left (Like 10, 5, 2.5, 1.25, 0.125, etc.). Because when you halve repeatedly you ...
0
votes
1answer
50 views

Integrals with infinite bounds sometimes written as limits, sometimes not?

When I saw Wikipedia's notation for the inverse Laplace transform, I became curious if there was a reason behind it. Is there a reason why Wikipedia writes the inverse Laplace transform as this ...
1
vote
1answer
52 views

Is the set (0, $\infty$) open?

A set is open if it doesn't contain any of its boundary points. I think 0 is a boundary point here and I think it's the only one. So is the set open?
0
votes
1answer
209 views

Finding a limit using arithmetic over cardinals

What is the value of: $$\lim_{n \to \infty} \frac{n}{2^n} (n \in \mathbb{N})$$ It seems to me that I can use L'Hopital's rule, but does that rule take into account types of infinity? More precisely, ...
4
votes
1answer
155 views

Infinite square-rooting

$ \lim_{n\to\infty} {\sqrt{1+{\sqrt{2+{\sqrt{\cdots +\sqrt{n}\ }\ }\ }\ }\ \ }\ } = ? $ Either closed answer or an upper bound would help.
0
votes
1answer
46 views

Evaluation of a limit

Here is a question on limits. I would like to ask help. Here it goes: $$\lim_{N\to\infty}\left(\frac{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n+1}}{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n}}\right)$$ ...
2
votes
3answers
251 views

Why can't you count real numbers this way?

Sorry but this is probably a naive question. Why can't you generate real numbers by a*10^b, the same way as rational numbers by a/b? a and b could be integers so that you would start counting real ...
1
vote
2answers
35 views

Which sign does $\lim_{n\to\infty}(-2)^n$ have?

How can I express $\lim_{n\to\infty}(-2)^n$ using $\infty$? Which sign does it have, plus or minus?
31
votes
7answers
1k 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 ...
1
vote
2answers
659 views

Limits of trigonometric functions as $x$ approaches $\infty$

A while back I ran into a problem in which I had to analyze the graph of $f(x) = ( \arctan x )^2$. I was fine until I had to evaluate the limit of the function as is approaches infinity to determine ...
0
votes
4answers
310 views

Why does Wolfram Alpha state that $-\infty/0 = +\infty$?

I ran into a scenario when practicing L'Hôpital's rule which yielded -infinity/0. I broke this down into $-1 \cdot \infty \cdot \frac 1 0$, which I assumed equaled $-1\cdot\infty\cdot\infty$, which ...
1
vote
1answer
129 views

Indeterminate form in solving an integral

I have this integral, and the solution gives an indeterminate form for the value $\alpha = 1$, can you explain to me how to solve the indeterminate form? $$\int_{\beta}^{+\infty} x^{-\alpha} dx = ...
2
votes
1answer
532 views

Residue at infinity (complex analysis)

I have trouble with the residue : at $z = \infty$. I tried to solve it at $z=0$ but it turns out that I was wrong while $z=0$ is not a pole. I must solve it at $z=2$ but I'm stuck. Any suggestion ...
3
votes
4answers
114 views

Can someone help me solve this limits question?

$$\begin{align}\lim x → ∞\end{align}$$ $$\begin{align} f(x) = {\frac{2^{x+1}+{3^{x+1}}}{2^x + 3^x}} \\ \end{align}$$ I tried using L Hopitable but that gives the same expression. Also tried using ...
0
votes
1answer
232 views

Is this definition of limit at infinity of complex functions correct?

In my book (Churchill), a limit of a function at infinity is defined as: $$ \lim\limits_{z \to \infty}f(z) \equiv \lim\limits_{z \to 0}f(\frac{1}{z}) $$ But why can't you define the point at infinty ...
0
votes
1answer
65 views

Are there countably many infinities?

$\aleph_0$, $\aleph_1, \aleph_2$ and so on are indexed by a natural number so shouldn't there be countably many infinities?
2
votes
2answers
1k views

How to solve sigma summation from 1 to infinity

I was reading this paper and I came across this algorithm: on page 2, which I'd like to compute on a computer (programatically), but I don't know how you'd go about evaluating a summation to ...
2
votes
1answer
88 views

Show that there exist infinitely many primes of the form $6k-1$ [duplicate]

This is a question on the text book that i have no way to deal with. Can anyone help me? Show that there exist infinitely many primes of the form $6k-1$
3
votes
5answers
351 views

How to explain infinty to a $3^{rd}$ grader?

In my country in $3^{rd}$ grade in math kids learn the four basic arithmetic operation (addition, subtraction, multiplication and divison) up to $10 000$. My sister this year goes to $3^{rd}$ grade ...
0
votes
2answers
59 views

Why does this limit work?

Let $h(x)= (1+1/x)^x$ and $g(x)$ be another function. Now suppose $\lim\limits_{x \to \infty} g(x)= \infty$. Then $\lim\limits_{x \to \infty} h(g(x))$ =$\lim\limits_{x \to \infty} h(x)=e$. I would ...
4
votes
1answer
323 views

Hilarious Comic … DiffyQ and infinity ensue…

I ran across this comic, and it's gold. It is orginially published here If I am correct, the first panel alone defines a self-referential loop if not a differential Equation: $X$: Amount of Black ...
3
votes
3answers
202 views

How does the chain rule for limits work?

I have to evaluate the limit of this function, $$\lim_{x\to0^+} \arctan(\ln x)$$ I already know the answer, it's $-\dfrac{π}{2}$, but the only part I don't get it, how does it come to that? I did ...
10
votes
6answers
548 views

Difference between approaching and being exactly a number

When we take a limit, we say that the value is never equals that number, but approaches it, like in $$\lim_{n\to\infty}\frac{1}{n} = 0.$$ It never reaches $0$, but becomes closer and closer to $0$. ...
0
votes
2answers
128 views

Does infinite time = time with no end = never?

Say an object is to travel from point A to point B, a finite distance of 2 meters. Say the object travels at 1m/s. After 1 meter the speed of the object is halved. After another half of the previous ...
0
votes
1answer
111 views

Limit of variable to zero multiplied with infinity

I was attending a lecture in computational fluid dynamics when an equation popped up with a variable that could go to infinity. My mind wandered and I started thinking of the following, completely ...