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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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 ...
0
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1answer
259 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 ...
0
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2answers
72 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 ...
0
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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 ...
1
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1answer
184 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 ...
1
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1answer
122 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 ...
0
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1answer
52 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
2k 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 ...
0
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3answers
113 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" $x$ ...
2
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5answers
255 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.
0
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2answers
110 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 ...
3
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0answers
167 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 ...
1
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1answer
44 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 ...
0
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2answers
66 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 ...
2
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2answers
50 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
54 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
105 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. ...
6
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3answers
702 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 ...
11
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4answers
441 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
175 views

Does infinite equal infinite?

I have a question. Let $x$ be infinite. $$2x=\infty\times2, \quad 2x=\infty$$ So actually, does $2x=x$?
1
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1answer
80 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
165 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 ...
0
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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 = ...
3
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6answers
559 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 ...
0
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0answers
149 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 ...
0
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2answers
39 views

Countably or Uncountably Many Discontinuities

I want to know why the following function has uncountably many discontinuities: $$f(x)=\left\{\begin{array} & x^2 & x \not \in \mathbb{Q} \\ 0 & \text{otherwise} \end{array}\right .$$ ...
0
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1answer
117 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 = ...
0
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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
4k views

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 ...
3
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1answer
967 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
424 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 ...
4
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4answers
5k views

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 ...
2
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3answers
122 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 ...
4
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2answers
176 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 ...
3
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1answer
69 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 ...
15
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3answers
599 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 ...
2
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1answer
57 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 ...
2
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1answer
138 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 ...
0
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1answer
43 views

Laurent Seies and Res

Prove that for any Laurent series f(t) one has "Res(f') = 0"? I know for a Laurent series of a complex function f is a representation of that function as a power series which includes terms of ...
3
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4answers
149 views

random thought: are some infinite sets larger than other [duplicate]

I was in the shower today and I just thought of this so I'm asking it. I'm sure this has been thought of before. Let's say we have two sets, the set of all even numbers and the set of all natural ...
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3answers
75 views

How to evaluate this limit with l'hopital's rule

is it possible to use L'hopital for this or is there another method I'm missing? I have no idea how to even start this. $$\lim_{x\to \infty} \frac{(9x+1)^\frac12}{x+1} $$
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0answers
50 views

Number of real numbers between 0 and 1 vs number of all integers [duplicate]

Let a be the number of real numbers between 0 and 1 Let b be the number of all integers. Can I say ...
2
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3answers
151 views

many infinities, how many zeroes? [closed]

although i am aware of the term non-standard analysis i have, as yet, no clear idea what it signifies. but i have often wondered about the pseudo-equation $$ \frac1{\infty} = 0 $$ which one may ...
0
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1answer
36 views

information content of a quadratic surd

how much information is required to construct the equation: $$ X^2 - 2=0 \; ? $$ suppose, in a spirit of seasonal festivity, we squander a few further bits, and pamper ourselves with the additional ...
0
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2answers
29 views

Infinite expansion of non-linear expressions with 3 or more variables

I just realised that if we expand any of the non-linear expression with power of 3 or more we can't stop expanding them until we are dead. So for example: Expansion: $(a+b+c)^2 = a^2 + b^2 + c^2 + ...
2
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0answers
45 views

Cantor, longish lines and the Landau -o notations

in general terms this question is about the behaviour of functions of a real variable as their argument $\rightarrow \infty$. i will present the matter as concisely as i can, but my presentation will ...
0
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0answers
48 views

Number of ways to cut a square

How many ways are there to cut the unit square into two pieces? And how many ways are there if the two pieces must have equal area? Some special cases: A. If the cut is required to be a horizontal ...
2
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0answers
38 views

Fine-grained way to measure infinity

It is known that the cardinality of $R$ is equal to the cardinality of $R^2$, $R^3$, etc. But, intuitively these sets have different sizes. A possible way to formalize this intuition is to talk about ...
0
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0answers
42 views

A simple question on limits

Is it true that $$ \lim_{x\to+\infty} \mathbb{I}_{S=\{z\mid e^{-z}>0, z\in\mathbb{R}\}}(x) = 1,$$ where $\mathbb{I}_{S}(x)$ is an indicator function for $x\in S$?
2
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2answers
59 views

Is probability meaningful in cases of infinity?

Is it meaningful to speak of probability in cases of infinity? For instance, consider me having an infinite line of balls arranged in the manner: - Red, Green, Blue, Red, Green, Blue, Red....... ...