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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Limit when an expoent goes to infinity

Please could someone help me and see if my solutions are correct for these two limits Let $n \in \mathbb{N}$ and $y \in \mathbb{R}$ and $y>0$. Case 1 $$\lim_{y \to \infty} ...
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1answer
27 views

Number of k-permutations that have odd number of an element

I want to find a recurrence relation $h_k$ for the number of k-permutations of $\{\infty a,\infty b, \infty c, \infty d \}$ that have an odd number of a's. I let $h_0=0$ because there is no odd ...
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6answers
2k views

Why is there antagonism towards extended real numbers?

In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept ...
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3answers
77 views

Why is the integral of sec^2(x) from 0 to pi infinity?

Why is it, if you take the integral of sec^2(x) from 0 to pi, my calculator returns "infinity" as the answer, but according to the second fundamental theorem of calculus, I got 0 with my own work. I ...
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2answers
70 views

Is this true that if limit approaches infinity the function equals to zero?

I would like to know if the following is true. If $$\lim_{z\to \infty} 1/f(z) = \infty$$ is that equivalent to $$\lim_{z\to \infty} f(z) = 0?$$
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2answers
26 views

Questions about hyperbolas and integration

I have a couple of questions regarding hyperbolas and their integrals. If it's too much, don't feel like you have to answer all 3 questions. My first question: The integral of a function like 1/x^2 ...
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11answers
2k views

Smallest next real number after an integer

This might be a silly question, but is it possible at all for n.00000...[infinite zeros]...1 to be the next real number after n? If not, why not? Firstly, I know (I think) that $$\lim_{x\to \infty} ...
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1answer
46 views

Supposing time were infinitely divisible; why are two minutes and one minute not of equal duration? [closed]

If time were infinitely divisible then an infinite number of 'seconds' would comprise both one minute and two minutes. But one minute does not equal two minutes. So, why are two minutes and one minute ...
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4answers
143 views

Does $\infty^0=1$?

I was wondering if $\infty^0=1$. Some people have told me that there is no answer; it is undefined. Others have told me that the answer is $1$, using the rule $a^0=1, \ a\neq 0$. If it is truly ...
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0answers
23 views

Infinite One-Time Pad

As you know, when used correctly, a one-time pad allows one to send a message, such that the only thing that can be found out about it is the maximum size (which is also the key length.) It is ...
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2answers
38 views

Evaluating a limit as $x\to -\infty$

I am trying to evaluate $$ \lim_{x \to -\infty} \left(1+ \frac{1}{x}\right)^{x²}. $$ I'd say it tends to 0, 1 or something linked to $e$ but I have no clue how to prove this... I'm getting really ...
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3answers
32 views

Computing the limit of this function

So I have an improper integral: $$ \int_0^\infty \frac{13x}{x^2+1}-\frac{65}{5x+1} dx $$ I have solved the integral into this: $$ \lim_{t \to \infty} ...
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5answers
2k views

Is half a pie as big as a whole pie?

I am reading an e-book called To Infinity and Beyond by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole". In ...
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1answer
107 views

What is zero times infinity? [duplicate]

If any number times zero is zero and any number time infinity is infinity, then what do you get when you multiply zero times infinity? Do they cancel one another out and equal any number since any ...
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2answers
65 views

How to show by the Root Test that $\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$ converges or diverges

How do I show by the Root Test that $$\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$$ converges or diverges? This is what I have done so far. Since we take $\sum\limits_{i=1}^\infty \sqrt[n]{|a_n|}$, we ...
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2answers
39 views

What this statement is really saying to prove one Real number has missed the bijection with Integers?

In a Combinatorics text, I find this: Not all infinite sets have the same cardinality. Consider the set of all integers and the set of all reals. Assume that the set of reals can be put in ...
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1answer
87 views

Applications of infinity in real life [duplicate]

I am writing a mathematical essay and would like to focus on the concept of infinity. I am not sure of any real life applications of infinity to write about or some way to narrow down the topics. Does ...
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1answer
91 views

Are there any infinites not from a powerset of the natural numbers?

With the cardinality of the natural numbers as $|\mathbb{N}| = \aleph_0$ and its powerset as $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the continuum hypothesis and the axiom of choice says that ...
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1answer
37 views

infinity between two points on a line

I remember from school that the number of points on a section of a line is infinite. On the other hand, when you reach the number two in a number sequence, that is a number and how big the number is, ...
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5answers
1k views

how do we assume there is infinity?

Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets. We know that our universe doesn't contain infinite number of elements, so how do we assume there is ...
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3answers
101 views

Interpretations of $\frac{\infty}{\infty}$

I am trying to understand the physical sense of the mathematical construct $\frac{\infty}{\infty}$ Suppose we have a function $f(x)$ representing some physical construct depending on a "quantity" $x$ ...
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5answers
196 views

Limit of $(n-k)! \cdot n^k$ as $n$ approaches infinity

Is it true that $(n-k)! \cdot n^k$ tends to $n!$ as $n \to \infty$? I think it is correct but can't think of a satisfying proof.
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2answers
95 views

Does sum of all natural numbers contradict another rule?

I must say that I am not a mathematician, just a enthusiast who likes to read all the "weird" results in mathematics. I read that sum of all natural number equals to $-1/12$ and I am also aware that ...
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0answers
60 views

Limits of infinite processes that terminate in finite time - checking my understanding?

I am a computer scientist by training, but have a fair amount of math background that I've picked up through classes, teaching, and general interest. A student of mine posed a question to me. I think ...
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1answer
35 views

A function that escapes to infinity with a finite input

I was wondering whether there exists a function that escapes to infinity with a finite input. For a specific example, how about $f(0)=0$ and as $x$ tends to $10$, $f(x)$ tends to infinity. The use of ...
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2answers
58 views

Evaluating a limit with infinity

I'm taking the limit as x approaches infinity from the left (-) of: $$ \sqrt{x^2+2x}- \sqrt{x^2-2x} $$ However I'm not sure how to go about this. I'm at: $$ \sqrt{ \frac{x^3+4x^2}{x+2x}}- \sqrt ...
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2answers
37 views

Finding asymptotes of $(2-x^2)/(e^x)$

I was trying to solve some exam question on calculus 1, and i found this "Sketch the graph of $(2-x^2)/(e^x)$" I'm interested to find Horizontal Asymptotes of the graph. 1) when x approaches ...
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2answers
40 views

Can real number infinity be bigger than other real number infinity?

I know that 2 countable infinities are considered equal because you can pair each element in one set two an element in another one. But, for example, if we let all real numbers between 3 and 5 be ...
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2answers
76 views

If the Kleene star of countable sets is countable, how are the real numbers uncountable?

The formal languages we use to represent number systems are interchangeable, which is why we don't hesitate to use different notations, e.g. hexadecimal, octal, binary, etc... to represent the reals. ...
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3answers
675 views

Limits at infinity

I'm working with limits at infinity and stumbled upon this exercise where I want to evaluate the indicated limit: $$\lim_{x \to \infty} \frac{1}{\sqrt{x^2-2x}-x}$$ I tried to solve it by doing the ...
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4answers
421 views

Partitioning the naturals into an infinite number of large sets

Is it possible to partition the positive integers into an infinite number of disjoint large sets ?
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5answers
153 views

Does infinite equal infinite?

I have a question. Let $x$ be infinite. $$2x=\infty\times2, \quad 2x=\infty$$ So actually, does $2x=x$?
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1answer
60 views

Function with a constant infinite order derivative, infinite final value, 0 initial value, and graph that resembles geometric growth

Please forgive my vocabulary & usage because I'm only a math amateur, so I'll try to describe this the best I can. Does such a function exist that has an infinite order derivative with a constant ...
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3answers
141 views

Why are irrational numbers uncountable and rationals contable?

Question 1: Why are irrational numbers uncountable and rationals contable? I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers ...
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1answer
26 views

Probability on the plane

Problem. On the Cartesian plane with origin O and x- y-axes, I randomly pick a point P. What is the probability that the line segment OP has a slope at least 1? Is the answer 1/4 or 1/2? answer = ...
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6answers
426 views

why does commutativity of addition fail for infinite sums?

While discussing the sum of a particular series, $\sum\limits_{n=0}^{\infty}{\left(-1\right)}^n$ (a sum that I've heard is alleged to be equal to $\frac{1}{2}$), it was mentioned to me that addition ...
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0answers
120 views

The sum of all the natural numbers [duplicate]

I've watched this video: ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12 Now, I'm not quite familiar with infinite groups and such, but common sense says that claiming that the sum of all natural numbers ...
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2answers
32 views

Countably or Uncountably Many Discontinuities

I want to know why the following function has uncountably many discontinuities: $$f(x)=\left\{\begin{array} & x^2 & x \not \in \mathbb{Q} \\ 0 & \text{otherwise} \end{array}\right .$$ ...
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1answer
83 views

What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series (An = 1 + 1/2 + ... + 1/n) and this natural logarithm L = ...
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1answer
38 views

map sum of square integers to a contiguous range of integers

Given a list $a$ of integers, $$n_{a_1}, n_{a_2}, ..., n_{a_d}$$ have $$N_a = \sum_{j=1}^d n_{a_j}^2.$$ The various $N_a$, $N_b$ etc. are integers, but are not contiguous: for example, if $d=2$, ...
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6answers
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Are the integers closed under addition… really?

Okay so I'm a 3rd year undergraduate studying Mathematics. I've proved in group theory countless times that the integers are closed under addition. It's obvious to me that they are. However this has ...
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1answer
888 views

Is there any mathematical or physical situations that $1+2+3+\ldots\infty=-\frac{1}{12}$ shows itself? [duplicate]

I just saw the proof that $$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts. I completely understood the proof and my question is NOT actually about the proof itself. At the end of the proof, ...
2
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0answers
341 views

An intuitive reasoning for 1+2+3+4+5… + ∞ = -1/12? [duplicate]

I was just watching this video: http://www.youtube.com/watch?v=w-I6XTVZXww In it, a professor working at the Nottingham University( Dr. Ed Copeland I think) shows how 1+2+3+4+5....+ ∞ = -1/12 Is this ...
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4answers
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How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate]

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have ...
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3answers
88 views

Number of Infinities in complex numbers

How many infinities would be there for complex numbers? Like there are 2 infinities (+infinity and -infinity) for the real numbers, is there a way to prove the number of infinities in the complex ...
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2answers
103 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
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1answer
68 views

Set theory, show a set is countable, homework. check my answer

I solved this question but there is something strange going on and I am unsure of myself. Would like someone to review it. We are given a total order (or linear order) $<^{*}$on group $A$ such ...
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3answers
552 views

What good is infinity?

I am becoming increasingly convinced that Wildberger's views are, if a little bizarre, at least not hopelessly inconsistent. When I was reading the comments in the video following (MF17), somebody ...
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1answer
53 views

Credit Given - Geometricly Modeling Infinity with 3 planes and 9 circles - Ratio of Circles

Refer to the attached diagram sketch to help visualize the equation. I am requesting help with an interesting math problem. Basically, I am diagraming infinity using three planes. These planes ...
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1answer
97 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...