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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88 views

Is it valid to write $\displaystyle \lim_{x \to 0} \frac{1}{x^2} = \infty$?

AFAIK the limes of a term does not exist if that term does not converge, but I haven't found a suiting question here yet. This probably is a double of a similar question.
3
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5answers
237 views

Product of all primes

Is the product of all primes a natural number? In other words, is this true: $$ \prod\limits_{\text{primes}} p_i \in \mathbb{N} $$ And if so, what about just some of them: $$ ...
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1answer
127 views

How useful is infinity? [closed]

If infinity in one case is just something that cannot be capped. Does it really find its use in something? Speed is a number and when Dr.Math can assume it can be infinite, in reality universe even ...
4
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1answer
202 views

Are there different types of infinity? [duplicate]

Today in class my professor mentioned that there are different types of infinity. This confused me at first because I always thought infinity is just infinity. What are the different types of ...
3
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4answers
106 views

Limit for $\lim _{n \to \infty}(n+2)^{2}\sin\frac{1}{n}$

Can't prove the limit $$\lim_{n \to \infty}(n+2)^{2}\sin\frac{1}{n}=\infty.$$ by definition it should start: Let $M>0$. There exists an $N>0$ for every $n>N$: ...
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2answers
234 views

On the Continuum Hypothesis

Let me start out by saying that I am not a mathematician. I read an article over at Scientific American that discussed the Continuum Hypothesis. I developed the following thought experiment that would ...
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5answers
104 views

Stuck with infinities

I have heard this "some infinities are bigger than others" . How can this be ? The context was that the cardinality of the set of integers is less than that of the cardinality of th real numbers , ...
1
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1answer
59 views

Question about $f$ continuous function with these conditions?

Suppose I have a differentiable and bounded function $$f: [0, + \infty) \longrightarrow \mathbb{R}$$ such that $$\forall x \in [0, + \infty) \, : f(x) \cdot f'(x) > \sin x.$$ The question is: ...
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1answer
47 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 ...
1
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1answer
135 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 ...
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2answers
112 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
63 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 ...
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1answer
101 views

Aren't two infinite 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 irregular, maybe in the sense of cubic planar version of Rado's graph: Every possible ...
6
votes
3answers
283 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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1answer
176 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
224 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
106 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
152 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
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2answers
169 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
53 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
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 ...
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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 ...
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3answers
108 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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2answers
124 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?
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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?
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2answers
105 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
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.
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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?
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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
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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
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1answer
49 views

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

How do you prove: $\mathop {\lim }\limits_{x \to \pm \infty } {a \over x} = 0$
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2answers
275 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
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
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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, ...
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3answers
72 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
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.
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1answer
59 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
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1answer
66 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
98 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
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 ...
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1answer
74 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
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1answer
211 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
159 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
49 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
255 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
36 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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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 ...
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2answers
680 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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4answers
315 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
130 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 = ...