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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Find the value of the Infinite product in terms of k which is a positive integer

$$\prod_{n=k+1}^{+\infty}\left(1-\frac{k^2}{n^2}\right)$$ The only help we have been able to find is that of Euler, anything would be amazing!
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Infinite product of negative numbers? $-1\times -1\times-1\times -1\dots=$ [closed]

Edited: Making the question as brief as possible to avoid future confusion and misunderstanding. Note This was moved as a separate question from: Product of all real numbers in a given interval $[...
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1answer
92 views

Why can't we keep adding axioms forever?

Let F be a formal system falling prey to Gödel's incompleteness theorems, implyng there is a true but unprovable statement, call it $G_1$. Of course, adding $G_1$ to the axioms of F doesn't solve the ...
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3answers
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Definition of limit as $x\rightarrow \infty$

Every time i get confused with the definition of $\lim_{x\rightarrow \infty}f(x)=L$. I could not find a reference that will give the definition. I am trying to write what i understood. See if this is ...
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2answers
49 views

Show that if |f(x)| converges in infinity, so is f(x).

I think that in a I should compare the function |f(x) - f(x) and 2|f(x)| but I am not sure how i would do that. Also, I am not sure how i should duduce what i want to deduce in b after i find a.
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339 views

Product of all real numbers in a given interval $[n,m]$

READ-ME I have now what I can call for myself answers to all my problems and subquestions proposed in this post, thus I accepted Strings answer as the answer to this question since it was of most ...
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1answer
16 views

How to compute log likelihood for impossible events?

I am defining a set $\mathbf{Z} = [p,q,r,s]$ such that $Pr(p)+Pr(q)+Pr(r)+Pr(s)=1$. Likelihoods are defined as follows \begin{align} \lambda_p&=\log \frac{Pr(x=p)}{Pr(x=s)}, \hspace{2mm} \...
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0answers
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How do we accomplish the subtraction of two infinities in a PWL Approximation?

I am trying to implement a piecewise-linear function of an M/M/1 Queueing system in an ILP to approximate the delay values. I have expressed my PWL constraint as follows: $\alpha_{i}+ \beta_{i}u_{n_s} ...
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3answers
830 views

Would an infinite random sequence of real numbers contain repetitions?

If random real numbers are selected from the set of all real numbers, for an infinite number of iterations, what is the likelihood of repetitions occurring?
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1answer
29 views

Finding percentage when infinity is involved

Is it possible to convert a function of the form $f(x)=ax/(a-x)$ to a form where you can find $f(x)/f(a)$? I'd like to find the percentage of $f(a)$ for $f(x)$ but this seems impossible while $f(a)$ ...
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2answers
24 views

Dichotomy in the number of regions on a plane formed by an infinite number of lines

I'm reading Knuth's Concrete Mathematics and we are dealing with recurrence relations. He proves that the number of regions $L_n$ formed by $n$ lines on a plane is $L_n=\frac{n(n+1)}{2}$. I don't ...
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2answers
70 views

Concept of infinity: Infinity - Infinity

What solution does $\int_0^\infty 1 dx - \int_a^\infty 1 dx $? yield if (i) $a\in (0, \infty)$ and (ii) $a=0$? From the Continuum hypothesis I concluded that each integral is uncountable infinite ...
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2answers
122 views

How does it equal -1/12? [duplicate]

So all my friends keep telling me that if you add up all the numbers from 1 to infinity, (1+2+3+4...) then the answer is -1/12. They showed me this proof with infinite sums, but I didn't understand it,...
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3answers
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Why is cardinality of set of even numbers = set of whole numbers?

I recently watched a YouTube video on Banach-Tarski theorem (or, paradox). In it, the presenter builds the proof of the theorem on the basis of a non-intuitve assertion that there as as many even ...
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0answers
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Let $f, g$ be defined on $(a,\infty)$ and $\lim_{x \to \infty}f(x)=L$ and $\lim_{x \to \infty}g(x)=\infty$, then $\lim_{x \to \infty}f(g(x)) = L$

If $\lim_{x \to \infty} g = \infty$, then for $M>0$, there exists $d_{1} >0$ such that if $x>d_{1}$, then $g(x) > M$. If $\lim_{x \to \infty} f = L$, then given $\epsilon>0$, there ...
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3answers
91 views

Finding $\sqrt {6+\sqrt{6+\sqrt{6+…+\sqrt 6}}}$

For positive integer $n$, $$x_n=\sqrt {6+\sqrt{6+\sqrt{6+...+\sqrt 6}}}$$ where $6$ is written $n$ times. How can we find the $x _\infty$ ? I coded a program any found that $x _\infty$ would be ...
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3answers
110 views

Is $\lim_{x \to -∞} (2+3x)^{2/3}$ positive or negative? [closed]

$\lim_{x \to -∞} (2+3x)^{2/3}$ Is this $(-∞)^{2/3} = (-∞^2)^{1/3} = +∞$ ?
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4answers
57 views

Finding limit without using $(a-b)(a+b)$ method

I'm working on this problem: $$\lim_{x\to \infty} (\sqrt {x^2 + 2x} - \sqrt {x^2 - 4x})$$ I tried the following approach and currently it's wrong: $$\sqrt {x^2 + 2x} - \sqrt {x^2 - 4x}$$ Taking out ...
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0answers
40 views

Does log(aleph-null) have any meaning?

I'm familiar w/ the meanings and derivations of $\aleph_0$ and the general consequences of the continuum hypothesis (and the discussions at this question. ) So, if it turns out that $2^{\aleph_0} = \...
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1answer
36 views

Is $\sum_{n=0}^\infty (a \cdot r^n)$ equivalent to $\lim_{n \to \infty}\sum_{k=0}^n (a \cdot r^k)$?

In other words, when writing down an infinite sum, are we always implying that it's actually the limit of that series as the number of terms approaches infinity, or is there some subtle difference?
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1answer
154 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 ...
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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 ...
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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 $x\...
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0answers
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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 ...
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4answers
89 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} {e^{-x}}...
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2answers
67 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
67 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
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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 $\frac{\infty}{\...
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2answers
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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 ...
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1answer
53 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
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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
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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.
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1answer
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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 ...
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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 (...
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0answers
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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 ...
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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?
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1answer
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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 ...
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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 ...
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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 $1/n$ ...
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260 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 ...
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1answer
42 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 ...
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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 ...
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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 ...
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3answers
98 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
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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 $10^{\aleph_0} > \beth_1$ ...
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1answer
34 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 ...
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1answer
83 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 ...
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2answers
106 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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596 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 ...