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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-5
votes
3answers
83 views

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$ [closed]

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$. How do you prove it?
2
votes
2answers
189 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
56 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
121 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?
101
votes
16answers
14k 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
68 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
56 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
503 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
62 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
61 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
105 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
142 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
79 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
87 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
186 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
58 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
89 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
227 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
222 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
52 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
446 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
44 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?
37
votes
9answers
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 ...
1
vote
2answers
2k 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
3answers
573 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
324 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
3k 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
119 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 ...
1
vote
1answer
669 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 ...
1
vote
1answer
87 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
2k 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
104 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 $6k-1$
4
votes
5answers
476 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
70 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 ...
6
votes
1answer
482 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
2k 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
736 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
238 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
278 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 ...
4
votes
2answers
976 views

Pi might contain all finite sets, can it also contain infinite sets?

In a previous, and quite popular, question it was discussed about whether or not $\pi$ contains all finite number combinations. Let us assume for a moment that $\pi$ does in fact contain all finite ...
0
votes
2answers
618 views

Question on limits and infinity

Just to clarify, the limit of $x \nearrow 0$ from the left of $1/x$, would be $-\infty$, and the limit of $x \searrow 0$ from the right of $1/x$, would be $+\infty$ right? This is only true when its ...
2
votes
1answer
202 views

More than one blocks of infinitely repeating digits in a number

TL;DR: Meaning of these types of numbers: $1.2\overline{34}5\overline{67}$; There exist (rational) numbers that are non-terminating, but have a repeating form of digits (e.g., $ 1.2\overline{34} ...
3
votes
2answers
131 views

How can evaluating the limit of function give a different result after rationalizing it?

One of the examples in Calculus: A complete course is finding $\lim_{x\to \infty} (\sqrt{x^2+x}-x)$. At first it seems to produce a meningless $\infty-\infty$, but by rationalizing it we eventually ...
1
vote
2answers
2k views

Must an infinite intersection of infinite sets be infinite?

If $A_2$ is a subset of $A_1$, $A_3$ is a subset of $A_2$, and this goes on infinitely and all contain an infinite number of elements, then is the intersection from $n=1$ to infinity, infinite as ...
0
votes
2answers
688 views

Sum from infinity to infinity

How does one evaluate the following limits? 1) $\lim_{n \rightarrow \infty} \sum_{k=n}^\infty 1$ 2) $\lim_{n \rightarrow \infty} \sum_{k=n}^\infty k^{-1}$ 3) $\lim_{n \rightarrow \infty} ...
1
vote
1answer
130 views

Construction of an infinite number type and other ideas

Construct an infinite number(?) that has a beginning, an infinite middle, and a end; such as 1000...0001, or 98111...1114 etc. Has this type of number been explored? Under some simple multiplications, ...
1
vote
1answer
1k views

Proper Subset of an Infinite Set is Equinumerous to the Set Containing It

I noticed that there is a question about $S$ being denumerable, which implies $S$ is equinumerous with a proper subset of itself, but what about an infinite set? That is, how to do I prove that every ...
5
votes
3answers
254 views

Are there “numbers” with infinite number of digits (to the left) and are they useful?

Are there "numbers" with infinite amount of digits (to the left) and are they useful?(not talking about p-adic numbers) By useful I mean used in math (or something) and not a dead end idea. I guess I ...
1
vote
1answer
321 views

What would happen if infinity was treated like a number?

Let's take the equation $2x = 1 + x$. We know that in the real numbers we can cancel out an $x$ on both sides (since the inverse of $x$ exists) and we get that $x=1$. However, if we include infinity ...
5
votes
1answer
799 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 ...