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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Indeterminate form limits question

$$\lim_{x\to 0}\frac{10^x - 2^x - 5 ^ x + 1 } {x\tan x} $$ This is an indeterminate limit. I want help in solving this problem. Thanks in advance
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1answer
205 views

Series of numbers $1^2 + 2^2 + \dots$ [closed]

If we have a series of numbers $1^2 + 2^2 + 3^2 + .. +(10^n)^2$ ~ 1/3 * ${({10}^n)^3}$ and if we have another series of numbers $1^{29} +2^{29} + 3^{29} + ..+ (10^n)^{29}$ ~ 1/30 * $(10^n)^{30}$. ...
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1answer
68 views

Countable and Uncountable sets

Is $\mathbb{N}\cup\{a\}$, for some $a\not\in\mathbb{N}$ countable or uncountable? $\mathbf{Attempt: }$ It is true that a set is countable if there exists an injective function $f : S → N$ from $S$ to ...
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0answers
22 views

infinite dimensional Cramer-Wold theorem

The Cramer-Wold theorem states that if every fixed linear combination of $d$ random variables converges to a normal distribution, then the $d$ variables jointly converges to a multivariate normal ...
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1answer
72 views

A Monkey Choosing Real Numbers for an Infinite Time

A common illustration of the nature of infinity is that, given an infinite amount of time, a monkey on a typewriter will, with probability $1$, produce the complete works of Shakespeare. Consider now ...
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1answer
135 views

Finding limit of a function as it approaches infinity

How do i solve the below without using L'hopital rule. The final answer obtained is $2/3$ ...
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2answers
103 views

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
35 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
192 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
90 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
34 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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10answers
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
53 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
146 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
26 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
36 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
158 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
69 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
152 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
113 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
44 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
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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
110 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
243 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
108 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
76 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
39 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
63 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
47 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
48 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
92 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
697 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
439 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
156 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
73 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
157 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
498 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
138 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
38 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
109 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
42 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
946 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, ...
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0answers
397 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 ...