# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$g(x)=x-1$$ ...
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### 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 perspective,...
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### 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?
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### 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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### 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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### Unique infinite subsets of the integers

Edit: Great points on the comments. There is no unique set of unique infinite subsets of the integers. Is this a better question? What is the largest possible cardinality of a set which is a set of ...
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### What is the largest set for which its set of self bijections is countable?

I recently came across a problem which required some knowledge about the self bijections of $\mathbb{N}$, and after looking up how to construct some different bijections I came across the result that ...
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### Limit to infinity and infinite logarithms?

When trying to evaluate$$\ln(\ln(\ln(\ln(\cdots\ln(x)\cdots))))$$I noticed that the answer was bound to be complex for any $x$. Plugging in a very, very large real number in for $x$ will eventually ...
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### Can I have something larger than infinite? [duplicate]

My question is "Can I have something larger than infinite?" Sometimes, we add infinite numbers into our set of numbers by simply extending our set and adding infinite numbers to it. But can't you ...
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### How small is infinite? [closed]

There are a lot of posts concerning how big infinite is, but I wonder how small infinite is. One can clearly see (ignoring a few things) that$$\frac{\infty}2=\infty$$Which means that no matter how ...
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### A box comprised of infinite number of small similar boxes.

On Wikipedia, I read, "A box can be thought of 'small boxes' infinitely repeating in all three dimensional directions" I don't understand what does Wikipedia wants to say with a box containing ...
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### The cardinality of Indra's net?

This question has been asked before, but the title of the post was so general that it received no sufficient answer. What is the cardinality of the set of jewels and reflected jewels in Indra's Net? ...
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### Classify the type of discontinuity at $x_0 = 0$

for (a) I think it is essential because the right side goes to infinity. for (b) I think it is removable because the function is not defined in $0,$ same goes for (c) I am really not sure about ...
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### What is the origin of the distinction between assignable and inassignable number?

Leibniz described his infinitesimals as being inassignable numbers in a number of texts, e.g., in his Cum Produisset that was analyzed in detail by H. Bos in a seminal text dating from the 1970s. The ...
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### Concept behind the limit to infinity?

I can across transfinite numbers and came up with a thought. What if$$\lim_{x\to\infty}f(x)=f(T)$$where $T$ was a transfinite number? Generally, in calculus, I have noted that it is two different ...
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### Prove the squared vector 2-norm is $\leq$ sum of 1-norm and infinity-norm

How do I prove that $$\|x\|_2^2 \leq \|x\|_1 \|x\|_\infty?$$
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### values that can be attained by random variables

Can a discrete random variables takes the values $+ \infty$ and $- \infty$ ? Can someone explain to me this with an example?
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### Can I assume the continum hypothesis in a proof

I am showing that the cantor ternary set has the same cardinality as $\mathbb{R}$ I want to use the fact that it is uncountably infinite and a subset of $\mathbb{R}$. ($|N| < |C| \leq \mathbb{R}$) ...