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
400 views

Limit to Infinity question?

$$\lim_{x\to\infty}\left(-\sqrt{-2x+x^2}+\sqrt{2x+x^2}\right)=2$$ I'm not sure how to go about solving this problem.
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2answers
66 views

Partial sum formula of a polynomial series?

I am trying to find the partial sum formula of the following series: $$ \sum_{y=1}^{\infty} \frac{4y^2-12y+9}{(y+3)(y+2)(y+1)y} $$ I have tried using Faulhaber's formula without success. I have also ...
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1answer
67 views

Is$\ \infty \times 0$ undefined in the extended real numbers?

And if it is, why? Is it a kind of postulate related to the fact that infinitely many points make a line?
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3answers
39 views

How can I evaluate this limit?

I'm studying for an upcoming midterm and i'm stuck on this question. It's asking me to evaluate the following limit and justify my answer. $\lim \limits_{x \to \infty} \sqrt{x^2+3x} - \sqrt{x^2-2x}$ ...
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1answer
35 views

What is the geometric way of relating zero to infinity?

I once saw (what I think was) a geometric way a relating zero to infinity. Something about a circle with radius 1 around the origin. Can you tell me where to find that? Thanks
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2answers
64 views

When is an infinite set larger than another infinite set?

Somewhat of a basic question that I've been pondering about, suppose we have 2 finite sets $A,B$, arbitrary sets with arbitrary elements that we know nothing about, except that they are both finite. ...
0
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3answers
50 views

The limit of $\sqrt{x^2+x+1}-\sqrt{x^2+1}$ as $x\to\infty$ [closed]

Currently I'm self studying limits. but I don't know how to get the answer to this question: $$\lim _ { x\to \infty }\left(\sqrt{x^2+x+1}-\sqrt{x^2+1}\right)$$ can someone help me
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1answer
31 views

What is the cardinality of all frames in time?

If we divide time into individual frames, then we would get a set of infinite frames. But what is the cardinality of such a set? Since time is continuous, like the real numbers, I would expect the ...
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0answers
57 views

Hilbert's hotel with uncountably infinite rooms: can you fit $\mathbb R^2$ guests?

I'm trying to expand on Hilbert's paradox. The original version states that: Suppose there is a hotel with a countable infinity of rooms (eg. $\mathbb N$), all of which are occupied. ...
0
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2answers
68 views

Limit that fail to exist

Does a limit that equals to infinity considered to exist ?? am confused !! for Example 1/(x-2)--> when evaluating the limit at 2 the result is 1/0 which is infinity while after looking at the graph ...
0
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1answer
67 views

Is$\ +\infty$ greater than any other number (surreal, superreal, hyperreal, …)?

Let$\ \mathbb{A}$ be an arbitrary totally ordered set and consider the largest element of the set of extended real numbers,$\ +\infty$. Can we say that$\ +\infty > \chi $, for *any*$\ \chi \in ...
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2answers
45 views

Question about $\lim_{x \to -\infty}\frac{\sqrt{10+11x^2}}{12+13x}$

$\lim_{x \to -\infty}\dfrac{\sqrt{10+11x^2}}{12+13x}$ = multiply top and bottom by $\dfrac{1}{x}=-\dfrac{1}{\sqrt{x^2}}$ My question is, why is the negative sign in front so crucial, I don't ...
0
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1answer
29 views

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$. My attempt: Let $x_n\to -\infty$ and $\epsilon\gt 0$. By the Archimedean Principle pick $N\in \mathbb N$ such that ...
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3answers
64 views

How do I solve $\lim_{x\to -\infty}(\sqrt{x^2 + x + 1} + x)$?

I'm having trouble finding this limit: $$\lim_{x\to -\infty}(\sqrt{x^2 + x + 1} + x)$$ I tried multiplying by the conjugate: $$\lim_{x\to -\infty}(\frac{\sqrt{x^2 + x + 1} + x}{1} \times ...
0
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3answers
69 views

Does this sequence diverge to ∞?

The sequence $(a_n)_{n \geq 1}$ is defined as follows: $$a_n:= \begin{cases} 0 \quad \text{if} \quad n \quad \text{is odd}\\ n \quad \text{if} \quad n \quad \text{is even}\end{cases} \quad .$$ Does ...
19
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7answers
3k views

There is no smallest infinity in calculus?

Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess. From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and ...
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3answers
225 views

Calculate exact value of and infinite sum [duplicate]

Im trying to find the exact value of the infinite sum : 3 + 1/3 + 1/27 + 1/243 + 1/2187 + ... I can see that to generate new terms we take the previous term and divide by 9 or multiply by 9. Not ...
2
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0answers
48 views

Is this set countably infinite or not?

"Far away, in the heavenly abode of the great god Indra, there is a wonderful net that has been hung by some artificer in such a manner that it stretches out infinitely in all directions. In ...
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0answers
52 views

Circles and the continuum hypothesis

I was trying to understand the undecidable nature of the continuum hypothesis and came up with the following question: The set of circles with a rational diameter is countably infinite (with ...
3
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3answers
133 views

Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...
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2answers
58 views

Limit approaching to negative infinity.

Q. Find $\lim _{x\to -\infty }\left(\frac{x^4\sin\frac{1}{x}+x^2}{1+|x|^3}\right)$ By inserting $x=-\frac{1}{y}$ and as $_{x\to \:-\infty \:}$ then $_{y\to \:0\:\:}$. By applying this my text arrive ...
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1answer
27 views

Help explain the set being constructed in this Cantor-Schroder-Berstein proof

The Cantor-Schroder-Bernstein theorem states that: Suppose $A$ and $B$ are sets. If $|A|\le |B|$ and $|B|\le |A|$, then $|A|=|B|$ Proof: So, $|A|\le|B|$ implies we can choose an injection ...
3
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1answer
81 views

Does there exist a connected 2-regular uncountable graph, or an uncountable path?

Does there exist a connected 2-regular uncountable graph? Can I use the axiom of choice to construct an uncountable path of elements from the reals? The question arose when reading this: Also, ...
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2answers
91 views

Sizes of infinity

I was just thinking about infinity (as you do) and thought the following. "There are infinitely many reals in the interval $x\in[0,1]$ and an 'equal number of reals' $x\in[1,2]$, so there are 'double ...
2
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2answers
44 views

Is there a thing named a “spiral plane” which is a plane but it's spiral?

Hello, I'm wondering if there is such thing like this. Is there a plane which is not flat but spiral and extending for infinity? I have drawn a representation for what I mean but it's not thorough ...
2
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8answers
612 views

Is 0.9 repeating = 1 disproved by asymptotes?

I'm discussing proofs that 0.9 repeating equals 1 with some friends, and they use asymptotes to disprove this. One says if we had the function $y=x/0.000\ldots1$ (and he's only using that impossible ...
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1answer
43 views

Degrees of freedom in each domain in Discrete, Continuous and Mixed Fourier Transforms

I'm having trouble with the different infinities involved in the Discrete and Continuous Fourier Transforms. In the DFT, we have a finite number $N$ time domain samples $x(i), 0\leq i<N$, which ...
2
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1answer
32 views

Help to prove this inductively defined function is surjective

Suppose that $A$ is a infinite subset of $\mathbb{Z}^+$. We construct a bijection $f:\mathbb{Z}^+ \rightarrow A$ and define $f(n)$ inductively as follows: Base case: Let $f(1)$ be the least element ...
2
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4answers
308 views

Can't find limit tending to infinity of a sequence

I'm stumped by $$\lim_{x \to \infty}\frac{1+3+5+\cdots+(2x-1)}{x+3} - x$$ My obvious first step was to get a lowest common denominator by $x(\frac{x+3}{x+3})$, giving $$\lim_{x \to ...
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0answers
38 views

Check proof of union of denumerable sets is denumerable too

I need to prove: If $A$ and $B$ are denumerable sets then so is their union $A\cup B$. In this case, denumerable is defined as: A set $X$ is said to be denumerable if there is a bijection ...
2
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1answer
69 views

The meaning of infinite series $\sum_{i=0}^\infty 2^{-i}$, its relation to partial sums and Cantor's diagonal argument

Let's define $S(n)$ as $S(n) = \sum_{i=0}^n 2^{-i}$. Obviously, $\lim_{n \to \infty} S(n) = 2$ and also $\forall n \in \mathbb{N}, S(n)<2$. Now my questions are about $Q = \sum_{i=0}^\infty ...
2
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1answer
94 views

Are $+\infty$ & $-\infty $ elements of the real number line? [duplicate]

Can someone give an explanation/proof of whether these two numbers lie on the real number line?
3
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3answers
88 views

How to prove that a set is infinite iff it is Dedekind infinite?

I need to prove the following: A set $X$ is infinite if and only if it is equipotent to a proper subset of itself Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or ...
3
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2answers
109 views

Is there any infinite quantity small enough to be affected by finite changes?

Hilbert's paradox of the Grand Hotel shows us, among other useful things, that the cardinality of any infinite set is a quantity equal to n more than itself for any finite n. I am interested in ...
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2answers
252 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [closed]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
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2answers
131 views

Bijection between an infinite set and its union of a countably infinite set

I have $A$ as an infinite set and $S$ as a countably infinite set, (so that means there exists a one-to-one correspondence between $S$ and $\mathbb{N}$). How do I show that there always exists a ...
1
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1answer
62 views

Why is a complex number plus infinity equal to infinity?

Why is $$2 + 3 i + \infty = \infty$$ according to Mathematica and Wolfram Alpha? Shouldn't it be: $$2 + 3 i + \infty = \infty + 3 i$$ ? After all: $$2 + 3 i + 10 = 12 + 3 i$$ and not: $$2 + 3 i ...
3
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1answer
159 views

how to prove : there are an infinite number of points on the circle

I think the follow problem is equal to the problem set 1.16.(a) in Principles of Mathematical Analysis (walter ruldin), And we take (a, b) in $R^2$, X in $R^i$ how to prove : there are an infinite ...
4
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2answers
642 views

Can the distance between 2 non-empty sets be infinite?

Intuitively I would immediately assume no, but that's not how things usually work in math and considering there are different kinds of infinities I haven't been able to find the answer. Here's my ...
2
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1answer
68 views

Method for computing limit of a sin function as x tends to zero

I have a question about computing $$ \lim_{x \to 0} \sin\left(\frac{\pi x}{4|x|}\right)$$ I found the limit of $\pi x$ and $4|x|$ seperately and ended with $\sin(\pi/4)$ which is equal to ...
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3answers
68 views

Method for computing limit of a function as $x$ tends to zero

I have a question about computing $$\lim_ {x \to 0} \dfrac{(2/x^3)+(1/x^2)+(1/x)+1}{(1/x^3)+1}.$$ I used a shortcut method of dividing by the highest power but I don't think that I can use this method ...
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4answers
138 views

May seem like a noob question: really, why can't we divide by 0? [duplicate]

Yes, I know, can't be answered, blah, blah, blah.... but here are a few of my theories. I know, plenty of other questions like this, but before marking this as a duplicate, consider this, my ...
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1answer
26 views

One set of functions larger than another set of functions?

This summer I've been slowly working through Halmos's Naive Set Theory. I'm not that far, but I know what lies ahead, which is proving that one infinite set is larger than another (the reals larger ...
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4answers
2k views

Can a set be infinite and bounded?

I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying : 'every infinite and bounded part of ...
3
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2answers
241 views

Teaching the Concept of Infinity to Children.

I was recently out with the family and we left it up to the children where we ate lunch (11 and 9 years old). They couldn't agree and were going back and forth calling each other names. This ...
2
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1answer
104 views

Is there a mathematical concept of fractions using transfinite numbers as numerators and denominators?

http://de.wikipedia.org/wiki/Cantors_erstes_Diagonalargument (German) http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument (English) While looking at Cantors method of proof, which he used to ...
8
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4answers
223 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
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1answer
38 views

Deffered annuity with perpetuity

An annuity immediate has $40$ initial quarterly payments of $20$ followed by perpetuity of quarterly payments of $25$ starting in the eleventh year. Find the present value at $4\% $ convertible ...
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1answer
57 views

Comparing density of countable infinite sets by examining the association

The two questions that i am asking are in bold. To be clear, i am talking about whole number here. Having seen 3 is everywhere by Numberphile that shows that almost 100% of the whole the numbers ...
0
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2answers
65 views

The limit of $(x^3+\cos x+e^{-2x})/(x^2 \sqrt{x^2+1})$ as $x\to\infty$

I have this infinity problem which I do not know the answer to: $$\lim_{x\to\infty}\frac{x^3+\cos x+e^{-2x}}{x^2 \sqrt{x^2+1}}$$ I thaught that because $x^3$ is the fastest growing part, this would ...