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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153 views

Could $\frac x0 = \pm\infty$? [duplicate]

Possible Duplicate: Is it wrong to tell children that 1/0 = NaN is incorrect, and should be ∞? I remember that dividing by zero is frowned upon, because it is said that there is no real ...
3
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1answer
429 views

Rational numbers and series going to infinity

(1) The sum of two rational numbers is a rational number. (2) The series $\sum\limits_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}$ is ...
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1answer
387 views

Why infinite sums of positive real constants definitely yield infinite?

According to the last step in proof of the unmeasurability of Vitali_set, it said that summing infinitely many copies of the constant $\lambda(V)$ yields either zero or infinity, according to whether ...
6
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1answer
423 views

Can an infinite cardinal number be a sum of two smaller cardinal number?

Let $\kappa$ be an infinite cardinal number. My question is whether there are $\lambda$ and $\mu$ such that both $<\kappa$ but $\lambda+\mu=\kappa$? If AC holds, then the answer is definitely ...
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1answer
106 views

Identifying an Error in Determining the Convergency of an Infinite Series

Given the infinite series of $(-1)^n/(nln(n))$ for $n = 2,3,4,\ldots$ to infinity, is the series conditionally convergent, absoultely convergent, or divergent? I took two approaches to solve this ...
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2answers
357 views

Harmonic Series Paradox

How to resolve the harmonic series paradox presented in this video by James Tanton?
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4answers
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Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$ [duplicate]

Possible Duplicate: Is value of $\pi = 4$? Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?
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4answers
1k views

Integration with infinity and exponential

How is $$\lim_{T\to\infty}\frac{1}T\int_{-T/2}^{T/2}e^{-2at}dt=\infty\;?$$ however my answer comes zero because putting limit in the expression, we get: $$\frac1\infty\left(-\frac1{2a}\right) ...
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4answers
312 views

Does $\log(x)$ stop at a finite value when x is infinite?

Does $\log(x)$ stop at a certain value when x is infinite? Or is it also infinite? I can see the graph go straighter and straighter in the horizontal direction, and I wonder if it will eventually be ...
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0answers
73 views

Simplifying this infinite series [duplicate]

Possible Duplicate: How can I evaluate $\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$ I have an infinite series like so: $$\sum_{i=0}^\infty (i+1)x^i$$ or basically $$ 1 + 2x + 3x^2 + 4x^3 +... ...
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1answer
114 views

Contour Infinites and Vector Spaces

We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
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0answers
112 views

How to observe infinity?

In my calculus course, there's example stated on the book: Given that $M$ is an ordered set and the sequence $\{a_n\}\subset M$, prove that there's a (weakly) monotonic subsequence of $\{a_n\}$. ...
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2answers
167 views

A vertical line in a cartesian coordinate system

Let's say I have points $A(a,a)$ and $B(a,0)$. What is the equation of the line $AB$? If I'm correct the slope is infinite, but it never has a y-intercept. This would give $y=\infty x$, but there are ...
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1answer
111 views

Why should the set have finite measure in the following proposition?

Here is a proposition in Royden: Assume $E$ has finite measure. Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise a.e. on $E$ to $f$ and $f$ is finite a.e. on $E$. ...
3
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1answer
232 views

Prove that a formal language is infinite

I'm having trouble with the following exercise: Let $\Sigma = \{a,b,c\}$ and $L$ be a formal language, that consists of all words which contain all three letters at least once. Show that $L$ is ...
3
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1answer
229 views

“Real” cardinality, say, $\aleph_\pi$?

Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? ...
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1answer
478 views

Continuity and pushing a limit inside the function's domain

Consider some right-continuous function $f:\mathbb{R\cup\{-\infty,\infty\}}\to [0,1]$. I have to evaluate (i) $\lim_{b \to 0^+} f(\frac{a}{b})$, and (ii) $\lim_{b \to 0^-} f(\frac{a}{b})$ where $a \in ...
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3answers
342 views

Find: $\lim_{n\to\infty} r^n$, for $r>1$ and $r<1$

Prove: $$\lim_{n\to\infty}r^n = +\infty\,, r > 1;$$ $$\lim_{n\to\infty}r^n = 0\,, 0 \le r < 1.$$ I am not quite sure how to prove this, but once someone proves it I will make sure to ask ...
2
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1answer
57 views

Is this set finite?

Let's say you are given a function $\mu:S\rightarrow(0,1]$ and you can additionally assume $$\sum_{s\in S}\mu(s)=1$$ Does this imply that $S$ is finite?
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2answers
239 views

How large is the infinity of real numbers [closed]

Umm ... Can someone disprove my proof that there are aleph-1 number of real numbers? Even comments to make my proof more rigorous are welcome. https://www.dropbox.com/sh/1fz28jlwrprh4jv/rhA7Ad7OtX
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2answers
302 views

What is the use of such concepts as potential infinity and actual infinity?

I'm aware of such mathematical concepts as and potential infinity and actual infinity. But I do not understand how those concepts are being used. Are there any applications to such concepts? Are there ...
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2answers
271 views

Is there such math concept as potential zero?

It seems that I need to use concept of potential zero in my work and I want to know whether I could reference some other works in order to fully understand what I'm dealing with. Specifically for my ...
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4answers
567 views

The concept of infinity

This evening I had a discussion with a friend of my about a mathematical riddle and the concept of 'infinite' The riddle Imagine a hotel with an infinite amount of rooms, and all of the rooms are ...
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6answers
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Is the set of all valid C++ programs countably infinite?

I have heard that the set of valid programs in a certain programming language is countably infinite. For instance, the set of all valid C++ programs is countably infinite. I don't understand why ...
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4answers
284 views

Limit of difference of two irrational functions

Firstly, this is not a homework. I just want to solve this limit for my own curiosity and self-learning. I have tried to solve this limit for 5-6 hours with no luck. Then I tried to read information ...
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5answers
200 views

Is infinite a infinite or finite

Long back I have watched a documentary based on a mathematician named Cantor. According to that documentary, Cantor claimed that infinite does not exists, it is only finite. Is that True?
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1answer
147 views

Normalize infinite range into finite one

First of all; I'm a programmer, not a mathematician so please excuse the informality of my math-vocabulary. I have a series of slopes, calculated out of random angles (their tangents). These angles ...
4
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1answer
187 views

What is the representation for a number that is not quite one?

If: $$0.\overline{9999999} \equiv 1$$ Then how would you represent a value that is infinitesimally close to one, but not quite one? i would have thought: $$1-\frac 1 \infty $$ But i would take that ...
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3answers
91 views

What is positive-0 squared minus positive-0?

I've got a basic limit problem that I think I'm solving the right way, but I've run into something that looks confusing enough to make me wonder if I'm doing it right. $$ \lim_{y\to0} \frac{1}{y^2-y} ...
0
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2answers
125 views

Is it possible to iterate through an infinite set?

Is it coherent to suggest that it is possible to iterate, one-by-one, through every single item in an infinite set? Some have suggested that it is possible to iterate (or count) completely through an ...
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4answers
163 views

Does every sequentially ordered infinite set contain sequentially ordered infinite subsets?

I am not very familiar with mathematical proofs, or the notation involved, so if it is possible to explain in 8th grade English (or thereabouts), I would really appreciate it. Since I may even be ...
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1answer
121 views

Countably infinite composition of injective functions.

Just out of curiosity that came from a topology homework assignment where I had to show the composition of 3 injective functions was injective. Suppose $f_i : A_i \mapsto A_{i+1}$ were injective ...
2
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1answer
8k views

Infinity matrix norm example

I have a brief question regarding the infinity matrix norm. The subordinate matrix infinity norm is defined as: $$\|A\|_{\infty} =\max_{1 \leq i \leq n}\sum_{j=1}^{n}|a_{ij}|.$$ This is derived ...
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1answer
571 views

A formal proof that a sum of infinite series is a series of a sum?

I feel confused when dealing with ininities of any kind. E.g. the next equation is confusing me. $$\displaystyle\sum^\infty_{n} (f_1(n) + f_2(n)) = \displaystyle\sum^\infty_{n_1=1} f_1(n_1) + ...
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4answers
769 views

Cardinality of the set of all functions from blank to blank

I'm trying to determine if a bunch of examples constitute countable or uncountable sets. One of which is the set of all functions from the positive integers to the positive integers. Word on the ...
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3answers
5k views

Is $\tan(\pi/2)$ undefined or infinity?

The way I have understood, $0/0$ is undefined or indeterminate because, if $c=0/0$ then $c\cdot 0=0$, where $c$ can be any finite number including $0$ itself. If we also observe a fraction $F=a/b$ ...
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4answers
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Is the statement “1/3 of the natural numbers are divisible by 3” true? Is anything similar to it true?

If we're talking about a finite set of the natural numbers, like those between 1 and 500 or 1 and a million, it seems to me that the fraction of numbers in that finite set that have a factor of 5 ...
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3answers
2k views

Comparison of two infinity [duplicate]

Possible Duplicate: Different kinds of infinities? Today I got to know that two infinity can be compared, But I want to know how is this possible? infinity will be infinity. If it doesn't ...
3
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4answers
975 views

Polynomial of degree $-\infty$?

I'm reading E.J Barbeau Polynomials. I'm in a page where he asks a polynomial of degree $-\infty$. Then I thought about $77x^{-\infty}+1$, but when I went for the answers, the answer to this question ...
2
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4answers
275 views

An infinite set having “one more element” than another infinite set

A classic example of homeomorphism is between a sphere missing one point and a plane To see this, place a sphere on the plane so that the sphere is tangent to the plane. Given any point in the plane, ...
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2answers
2k views

Why do the rationals, integers and naturals all have the same cardinality?

So I answered this question: Are all infinities equal? I believe my answer is correct, however one thing I couldn't explain fully, and which is bugging me, is why the rationals $\mathbb Q$, integers ...
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5answers
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Are all infinities equal?

A friend of mine was trying to explain to me how all infinities are equal. For example, they were saying that there are the same amount of numbers between $0$–$1$ as there are between $0$–$2$. The ...
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2answers
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Whats infinity divided by infinity?

This should be a simple question but i just want to make sure. I know from infinity/infinity is undefined. However if we have 2 equal infinities divided by each other it would be 1? And if we have ...
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1answer
1k views

Proving that there is no biggest number

Problem Elsewhere, I have seen it proposed by a mathematician that there is a "greatest number", as opposed to infinitely many numbers. This number is supposed to be exceedingly large, larger than ...
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7answers
4k views

Why do we say the harmonic series is divergent? [duplicate]

If we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we ...
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1answer
250 views

The relative rates of tending towards infinity of different functions?

In my reading, it says that the function $x/\log x$ approaches infinity slower than $x$ (I got that bit), but then it says that it also approaches it faster that the functions $x^{1-d}$, where $d$ is ...
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2answers
2k views

Less than infinity or Less or Equal to infinity

What is the difference between Less than infinity or Less or Equal to infinity? We cannot substitute infinity anyway, and have to use limits. So does it make sense to write less and equal to ...
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2answers
328 views

Proof that the Irrationals are Countable

Proof: Between any two irrationals lies a rational, by the Density of the rationals in the real number system. There are only countably many rationals; therefore, there are only countably many pairs ...
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3answers
291 views

What is wrong with this proof of the countability of the reals?

By using the following lemmas: A countable union of countable sets is countable. Cartesian products of integers are countable. Wouldn't it be possible to prove the countability of the reals ...
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1answer
384 views

infinity understanding problem?

between 0 meter -> 1 meter there are 100 cm. but each cm has infinite numbers : for example between 0..1 cm there are : ...