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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1answer
103 views

Reducing a double summation with infinite limits

I've been solving a Renewal theory problem and I end up with this function $m(t)=e^{-4t}\sum_{k=1}^{\infty}\sum_{i=2k}^{\infty}\frac{(4t)^i}{i!}$. How do I solve or reduce the double summation? Is it ...
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1answer
1k 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 ...
2
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2answers
205 views

How can we define infinitary proofs?

In the first order logic the usual notion of a formal proof for a sentence $\sigma$ from a theory $T$ is a "finite" sequence ($<\omega$ - sequeance) of sentences which each one of them is a valid ...
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1answer
124 views

n-Ball Volume and surface with $n \rightarrow \infty$

I am thinking about something I just read: The volume of the n-ball is given by $V_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac n 2 + 1)}r^n$ and its surface area is $S_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac ...
1
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1answer
135 views

Aren't two infinite asymmetric graphs always identical?

Suppose you have an infinite graph $G$. I assume $G$ to be cubic and planar. No further conditions, so it will be asymmetric, maybe in the sense of cubic planar version of Rado's graph: Every possible ...
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3answers
352 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? ...
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2answers
734 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, ...
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4answers
377 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 ...
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1answer
392 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) ...
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3answers
235 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 ...
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2answers
601 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
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1answer
68 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
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1answer
226 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 ...
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1answer
69 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 ...
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3answers
389 views

Infinite compact subset of $\mathbb{Q}$

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

Limit Involving Factorials

How would you go about calculating $$ \lim_{x \to \infty} \frac{x!}{(x - k)!} $$ for some constant $k > 0$?
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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.
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2answers
122 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?
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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"?
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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 ...
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1answer
56 views

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

How do you prove: $\mathop {\lim }\limits_{x \to \pm \infty } {a \over x} = 0$
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2answers
505 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 ...
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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+...) = ...
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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
106 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
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2answers
143 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.
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1answer
80 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
88 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 ...
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4answers
187 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
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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 ...
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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?
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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
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1answer
227 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.
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1answer
52 views

Evaluation of a limit

Here is a question on limits. I would like to ask help. Here it goes: $$\lim_{N\to\infty}\left(\frac{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n+1}}{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n}}\right)$$ ...
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3answers
459 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 ...
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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?
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12answers
3k 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 ...
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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 ...
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3answers
575 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 ...
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1answer
327 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 = ...
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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 ...
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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 ...
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1answer
686 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 ...
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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?
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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
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1answer
107 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 $6k-1$
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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 ...
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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 ...
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1answer
490 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 ...