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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How to prove that $0.01001100011100001111…$ is not periodic decimal number?

I have the following decimal number: $0.01001100011100001111...$ Notice how whenever we have one 0, we also have one 1, two 0's, two 1's, etc. How do you continue it to infinity and prove that this ...
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1answer
105 views
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168 views

Elaboration of infinite, finite and enumerable definition

I am starting to learn some of the basic concepts of math. The concept I am learning now is infinite, finite, and denumerable. I am having trouble understanding the book's definiton. I am hoping if ...
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3answers
278 views

Random Point on Infinite Line Paradox

I've invented a paradox, or at least I think I have. Here is how it goes: On an infinite line, a point is placed at random. You start at point 0 on the line, and your job is to find the point, but ...
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3answers
154 views

Is ∞ considered defined?

$\infty$ (Infinity) is not a number, but infinity is considered to be defined, right? There are expressions in mathematics such as: $\frac x0,0^0, \frac\infty\infty,$ which are not defined because ...
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3answers
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Does the concept of infinity have any practical applications?

I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world ...
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1answer
87 views

Are distinctions in definitions of “finite” material in, eg, topology or measure theory?

There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set: (Richard ...
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1answer
62 views

Please help me understand this formula for Fourier analysis

I'm a programmer with a poor knowledge of math. Could anyone tell me how to read the infinity above the sigma, and the n=1 below the sigma?
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5answers
2k views

How many different sizes of infinity are there?

It's pretty straightforward to say that there is an infinite number of different sizes of infinity, but then I thought, "What size of infinity is that?" My thoughts are that the number of unique ...
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1answer
112 views

Finding an element of the intersection of an infinite sequence of “compatible” sets of infinite sequences

Let $A$ be a set. Let $A^\omega$ denote the set of infinite sequences of members of $A$ (i.e., functions from $\omega$ to $A$). Define $\omega_n = \omega \setminus \{n\}$. Let $A^\omega_n$ denote the ...
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1answer
55 views

Comparing Improper Integrals Involving Infinity

From my current understanding: $K>J$ and $L>K$ , therefore $L>K>J$. How can I compare the first integral $I$ ?
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1answer
856 views

Comparison Theorem for Integral Calculus

I have narrowed it down to C, E, and F, since we know that $1/x^{1/5}$ is always greater than the original function for all $x\geq 1$. However, the second set of conditions is more difficult to ...
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0answers
102 views

Hadamard regularization isn't working out

As part of an exercise in a grad course on "mathematical methods" (always such a helpful name), I've been asked to evaluate $I=\int_0^{1/2}{(x^2-x+c)^{-2}dx}$ as a Hadamard finite part integral for $0 ...
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7answers
344 views

A proof of a property of limits

Today during lecture my lecturer showed us this property, but provided no proof. If $$\lim_{n\to\infty} {d_{n+1}\over d_n} >1$$ then $$\lim_{n\to\infty}d_{n}=\infty $$ Is this property legit? ...
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2answers
127 views

Why is this proof false?

I know this proof is false, but I don't know why. I need your help. The false proof says that it is possible to create a bijection between a subset of the rational numbers and the Power set of ...
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1answer
129 views

Infinite versus unendlich and double-negation

The German term for infinite is unendlich, which transliterates as non-ending, or non-finite. This is just word-play but from a constructive point of view, is the shift from a negative to a positive ...
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4answers
288 views

Is the powerset of every Dedekind-finite set Dedekind-finite?

Is the powerset of every Dedekind-finite set Dedekind-finite? I think this statement can be written in $\textbf{Set}$: If every mono (=injection) $f: A \to A$ is iso (=bijection), then every mono ...
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2answers
421 views

Infinite sum of floor functions

I need to compute this (convergent) sum $$\sum_{j=0}^\infty\left(j-2^k\left\lfloor\frac{j}{2^k}\right\rfloor\right)(1-\alpha)^j\alpha$$ But I have no idea how to get rid of the floor thing. I thought ...
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3answers
166 views

Finite sequences of unbounded value

If I have a finite sequence of expressions $a_1+a_2+a_3+....a_k=\infty$, does that imply that at least one such $a_j=\infty$? I know that if it didn't it would make the sum not equal to infinity, but ...
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What exactly is infinity?

On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and ...
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2answers
203 views

Is it viable to ask in an infinite set about the Cardinality?

Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?
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4answers
548 views

Does such a natural number exist, that it would be divisible by every other natural number

I've got to prove (or disprove) the following statement: $\exists x \in \mathbb{N} \; \forall y \in \mathbb{N}: y \mid x$, which translates into "It exists such $x$ from the set of natural numbers, ...
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3answers
755 views

Number of points on line segment

I know the line segment have a infinite number of points, but i know that exist different kinds of infinity ( $\aleph_0 $). My question is there same number of points on segment of line and entire ...
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3answers
173 views

What does the notion of different sizes of infinity really mean?

I have heard that there are infinities of various sizes. I was wondering what that actually means-how do we compare their cardinalities? I have just started real analysis and I am slowly coming to ...
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2answers
388 views

Multiples of numbers up to infinity

A question my wife and I were chatting about last night. Are there more multiples of 3 than there are of 17, if we count from 0 to infinity One point of view was since there are infinite multiples of ...
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1answer
71 views

Finite sums of infinite value

If a sum of a finte number of terms is infinite, does that imply that at least one term in the finite sum is also infinite?
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7 Drinks - 7 Flavors - Infinite variety?

Me and three friends are trying to find the answer to a question I posed about a self-service drinks machine in our local Burger King: There is a drinks machine that has 7 varieties of drinks (coke, ...
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2answers
118 views

Why do geometric sets such as $(\infty, x]$ never have infinity included?

I have a question about the use of infinity and geometric sets. Say I am trying to graph an equation, and the result is all values greater than or equal to, say, $3$. From what I've seen, the proper ...
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1answer
54 views

$A+\alpha\sim A$ when $\omega\le\alpha<h(A)$

I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$. If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
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2answers
128 views

Is there any Dedekind-infinite set can be split to two smaller Dedekind-infinite sets?

I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$. If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
4
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1answer
134 views

Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?

$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered. So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it ...
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Infinite expected value of a random variable

How can a positive random variable $X$ which never takes on the value $+\infty$, have expected value $\mathbb{E}[X] = +\infty$?
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133 views

Prove that if a set is Peano finite, then it is Dedekind finite.

I understand that this should be done by induction, but I have very limited knowledge on proof by induction. Could someone explain it in a way which also makes clear exactly what each stage of ...
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4answers
209 views

Prove sum is bounded

I have the following sum: $$ \sum\limits_{i=1}^n \binom{i}{i/2}p^\frac{i}{2}(1-p)^\frac{i}{2} $$ where $p<\frac{1}{2}$ I need to prove that this sum is bounded. i.e. it doesn't go to infinity ...
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3answers
154 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 ...
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1answer
444 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
408 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 ...
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1answer
443 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
107 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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373 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
2k 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
313 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
74 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
116 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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113 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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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 $E$. ...
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1answer
249 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 ...
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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? ...