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
151 views

Is this line of reasoning correct/valid?

I'm only in the second month of my first calculus course, so I'm not sure how much sense this question will make. I'll give it a try anyways though. Let's say you have the sum of an infinite series ...
0
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1answer
51 views

How can one prove this generalization?

In two dimensional space, the length of a vector is $$\sqrt{x^2+y^2}$$ In three dimensional space, the length of a vector is $$\sqrt{x^2+y^2+z^2}$$ How can one prove that in n th dimensional space ...
0
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1answer
34 views

A simple question about limits.

This may seem like a simple question, but I feel as if it is wrong but I am unsure why. Is it possible to evaluate a limit in two stages for example: say you know that $x(1- a)\rightarrow b$ as ...
1
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0answers
30 views

Applying a general function an infinite number of times

I am trying to learn more about infinite application of functions and functionals. My background is in quantum chemistry, so please forgive some of my notation and terminology. Motivation In ...
2
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4answers
88 views

Help finding $\lim_{x \to \infty} {(1 + e^x)}^{e^{-x}}$

$\lim\limits_{x \to \infty} {(1 + e^x)}^{e^{-x}}$ Here are the steps I have taken so far : $\ln{L} = \lim\limits_{x \to \infty} \ln{({1 + e^x})^{e^{-x}}}\\ \ln{L} = \lim\limits_{x \to \infty} ...
0
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2answers
66 views

Positive continuous function with non-zero limits in $\pm\infty$ whose integral over $\mathbb{R}$ is $1$?

Is it possible to create a positive continuous function with non-zero limits in $+\infty$ and $-\infty$ whose integral over $\mathbb{R}$ is $1$? I am studying the probability density functions and ...
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2answers
64 views

$\infty$ and $-\infty$ are to $\aleph_0$ / $\beth_0$ as “what” is to $\beth_1$?

So, I recently asked a question about whether $\beth_1$ had a negative, and I was promptly reprimanded because I confused $\aleph_0$ with $\infty$. Therefore, to help me understand the concepts ...
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3answers
33 views

Simplifying a $\lim_{x\to\infty}$ problem.

So I have a problem regarding limits in my calculus class: $$ \lim x\rightarrow\infty \frac {(1+2x^{1/6})^{2016}}{1+(2+(3+4x^6)^7)^8}$$ Basically what I've identified is that it's an ...
1
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2answers
39 views

limit of fraction with factorials

I am trying to take the limit of the following fraction : $$ \lim_{N \to\infty} \frac { N !}{(N-r)!} $$ Attempts : I tried using the Stirling approximation $\ln(n!) =n \ln n - n $ but I figured it ...
1
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1answer
50 views

Is the Cartesian product of two countably infinite sets also countably infinite?

I am trying to determine and prove whether the set of convergent sequences of prime numbers is countably or uncountably infinite. It is clear that such a sequence must 'terminate' with an infinite ...
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2answers
49 views

Infinite limit of trigonometric function

I'm trying to find the limit of a trigonometric function as x approaches $\infty$ so I can't use the fact that : $$\lim_{x\to \infty} \frac{1}{x} = 0$$ For example this limit : $$\lim_{x\to \infty} ...
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2answers
92 views

Proof of $+\infty=-\infty$ (Maybe)

I guess we can agree that $+0 = -0$. Now, after that, I was simply looking at some graphs. The graph of $\tan x$ shows asymptotes at x = $n\pi + \pi/2$. I got to thinking, what if they weren't ...
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3answers
73 views

What is $\lim\limits_{x \to \infty} \dfrac{x}{x-1}$? [closed]

I want to calculate the limit $$ \lim\limits_{x \to \infty} \dfrac{x}{x-1}. $$ I know that this can be achieved using l'Hospital but cannot figure out how to do this.
1
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1answer
74 views

Guests leaving Hilbert's Hotel?

I am a layman in this field so my understanding of the problem of "Hilbert's Hotel" is limited to the popular version presented to the public. We know that Hilbert's Hotel can accommodate any finite ...
0
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1answer
26 views

Infinite non deviating slope

Okay, I had a question that my math teacher didn't know the answer to, and that I haven't found an answer for on the web. Say you are graphing a system of equations, right, and you have ...
2
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0answers
20 views

How to count the number of rebound cycles of a ball in a 1d system after time t, where velocity doubles every rebound [closed]

This is something I dreampt up in a physics lab about imagining infinity but I never got round to modeling it. Seems like it could get out of hand pretty quickly! Imagine we have a ball in a 1d ...
12
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4answers
186 views

Prove that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$ [closed]

Using a graphing calculator, it seems that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$. How can this be proven?
2
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1answer
31 views

Approximating end behavior of a function by plugging in infinity

In Algebra 2, I learned to be able to tell if the end behavior of a function has an asymptote, approaches infinity, approaches zero, etc, by plugging in numbers closer and closer to infinity, or by ...
0
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1answer
36 views

Concept similar to extended real line in higher dimensions?

I have a question related to the notion of extended real line. I am a very beginner of this topic and in what follows I might say things that look no-sense for an expert in the field. The extended ...
23
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4answers
2k views

Is it irrational?

Suppose I generate a number $0 < x < 1$. In general, after the decimal point, the first digit is $1$, the second is $0$, the third is $1$, etc. However, every digit in position $n$ has a $1/n$ ...
1
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3answers
257 views

What is wrong with this infinite sum [closed]

We know that: https://www.youtube.com/watch?v=w-I6XTVZXww $$S=1+2+3+4+\cdots = -\frac{1}{12}$$ So multiplying each terms in the left hand side by $2$ gives: $$2S =2+4+6+8+\cdots = -\frac{1}{6}$$ This ...
1
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1answer
41 views

absolutely integrability implies function approaches zero at positive infinity

Is the following statement true? $$\text{If function $f$ is absolutely integrable on $[0, \infty)$, this implies } \lim_{x \rightarrow \infty} f (x) = 0.$$ If yes then how would I prove it? Note: I ...
0
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1answer
39 views

Interchange summation (outer infinite, inner dependent on outer)

For finite summation limits, I believe that the following holds (for some general function $f$): $\sum_{i=2}^n \sum_{j=1}^{i-1} f(i,j) = \sum_{j=1}^{n-1} \sum_{i=j+1}^{n} f(i,j)$ ... (1) However, I'm ...
0
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0answers
60 views

Why are positive rational numbers countable but real numbers are not? [duplicate]

If we can say that any positive rational number is countable or listable by showing that every positive rational number is the quotient of p/q of two positive ...
3
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3answers
86 views

Solution of the equation $\cot \theta = 2\cot 2\theta$

I've tried to solve the equation $\cot \theta = 2\cot 2\theta$ with the command 'Reduce' of Mathematica and obtained $\theta = n\pi$ as the solution with n an integer. But $\theta=n\pi$ is clearly a ...
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2answers
61 views

What is the result of a number greater than 2 raised to the power of {Aleph-0}?

So, I know that $2^{\aleph_0} = \beth_1$, right? What about another number, say $10$, raised to the power of $\aleph_0$? Is $10^{\aleph_0} = \beth_1$ also true, or is $10^{\aleph_0} > \beth_1$ ...
1
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1answer
33 views

What happens to Chebyshev polynomials integration when n=1

The integration of Chebyshev polynomials of the first kind has the following value, $$\int T_{n}(x) \, dx = \frac{1}{2} \, \left( \frac{T_{n+1}(x)}{n+1} - \frac{T_{n-1}(x)}{n-1} \right)$$ what happens ...
3
votes
1answer
80 views

Circle is similar to a polygon with infinite number of sides

It is know from the time of Euclid, that a circle is similar to a polygon with infinite number of sides. But this ^^ is informal. Do you know any formalization where it appears that a circle is a ...
6
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2answers
101 views

Why was $\aleph$ (aleph) chosen for infinities?

Why did Cantor choose a letter from the Hebrew alphabet to represent infinities, rather than using some Greek letter?
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3answers
592 views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
2
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1answer
77 views

Cardinality of polynomials with real coefficients

What is the cardinality of the set of all polynomials with real coefficients? I know the power set of R is "more infinite" than R, so to speak, but I'm unsure of how to prove that there does or does ...
3
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1answer
47 views

What is the derivative of $\int_{-10}^{-3} e^{\tan(t)} \,dt$ with respect to x?

We were learning about the Fundamental Theorem of Calculus today in my high school and the above integral came up as an example of an integral with a "constant" value. At first I accepted that the ...
0
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2answers
65 views

Different infinity, same limit?

I heard that there are different ranks of infinity, like $\aleph_0, \aleph_1, \aleph_2$, etc, my question is, the base of natural log, i.e. '$e$' is defined by a limit of taking $n\rightarrow$infinity ...
1
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1answer
49 views

What is an infinite gap minus another infinite gap?

I was asked this question on a quiz I received a few days ago and I was kind of confused on what the answer would be. Here it is, Set up and find the area between $$f(x)=x^2-x$$ and $$g(x)=x-1$$ ...
0
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1answer
35 views

How to disprove that $\text{ span }\{x_1,…,x_k\}=\text{ span }\{y_1,…,y_l\}$ if $x_i\in \text{ span }\{y_1,…y_l\}\ \forall i=1,…,k$?

If $y_1,...,y_l$ are vectors in vector space V and $x_i\in \text{ span }\{y_1,...y_l\}\ \forall i=1,...,k$, how to disprove that span$\{x_1,...,x_k\}=\text{ span }\{y_1,...,y_1\}$. In my ...
1
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3answers
104 views

How to solve this limit involving cube root and infinity?

How can I solve this limit? I know the answer is $2/3$. I tried factorisation, but solving the complicated denominator using L'Hopital's Rule returns a wrong answer, $0$. $$ \lim_{x\to\infty} ...
8
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3answers
970 views

The smallest infinity and the axiom of choice

The short version of this question is: which (natural) axiom should be added to ZF so that the statement "$\aleph_{0}$ is the smallest infinity" becomes true? A set $A$ is called infinite if it can ...
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2answers
65 views

How can a bijection be made from $\mathbb{N}$ to $\mathbb{Q}$ using diagonalization?

I'm studying Cantor's diagonalization, but something seems unclear to me. There is this table for diagonalization: ...
1
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1answer
43 views

Limit of a difference

Let $\lim_{n \to \infty} f_n(x) = f(x)$. Now consider $$\lim_{n \to \infty} (f_n(x) - f(x))$$ Usually I would say that $$\lim_{n \to \infty} (f_n(x) - f(x)) = \lim_{n \to \infty} f_n(x) - \lim_{n \to ...
2
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2answers
91 views

More numbers between $2$ and $4$ than between $2$ and $3$? (I am not a mathematician.) [duplicate]

Between $2$ and $3$ there are infinite numbers and between $2$ and $4$ there are infinite numbers. So which "infinity" is greater?
0
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1answer
189 views

Does Pi contain itself? [duplicate]

Alright, recently there was a question on 9gag whether the digits of $\pi$ may contain $\pi$ itself here's the original. One user had - in my opinion - a really plausible answer: Here's his answer. ...
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4answers
3k views

What's between the finite and the infinite?

I'm wondering if there are any non-standard theories (built upon ZFC with some axioms weakened or replaced) that make formal sense of hypothetical set-like objects whose "cardinality" is "in between" ...
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7answers
1k views

How can a Cauchy sequence converge to an irrational number?

I am a physics major and would like to clear a confusion regarding complete metric spaces. I am quoting the definition of a Cauchy sequence from wikipedia below Formally, given a metric space $(X, ...
0
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2answers
48 views

How come it be $\frac{3}{2}A$ and not only $A$?

OK I admit I was too lazy to type this question so I took a screenshot , I got it from the site @brilliant.org where it asked in terms of $A$ what would be the 2nd summation equation ? The explained ...
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1answer
36 views

Find the ratio of a geometric sequence such that its sum is $4$ times the first term

How to find the sum to infinity: the sum to infinity of a geometric progression is 4 times the first term. Find the common ratio.
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5answers
611 views

Is arithmetic with infinite numbers fictitious?

In 1933 Skolem constructed models for arithmetic containing infinite numbers. In a 1977 article Stillwell emphasized the constructive nature of Skolem's approach; see here. Is this at odds with ...
12
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1answer
424 views

Countable-infinity-to-one function

Are there continuous functions $f:I\to I$ such that $f^{-1}(\{x\})$ is countably infinite for every $x$? Here, $I=[0,1]$. The question "Infinity-to-one function" answers is similar but without the ...
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0answers
61 views

If we think of infinity as a number, how does it affect the compactness/completeness of a metric space?

I was recently reviewing some notes regarding compactness, in which the sequential definition is given i.e. "$A$ is compact if any sequence in $A$ has a subsequence which converges to a limit in $A$. ...
0
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0answers
37 views

If I can prove f(n) = g(n+1) by induction when n is finite, Can I prove f(n) = g(n) by taking n = $\infty$

I have to prove f(n) = g(n) when $n = \infty$. Now I can prove f(n) = g(n+1) by induction when n is finite. Can I say $f(n) = g(n)$ by taking $n = \infty$?
10
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1answer
424 views

Infinity-to-one function

Are there continuous functions $f:I\to S^2$ such that $f^{-1}(\{x\})$ is infinite for every $x\in S^2$? Here, $I=[0,1]$ and $S^2$ is the unit sphere. I have no idea how to do this. Note: This is ...