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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The limit of a rational function

$$ \lim _{x \to -\infty} \frac{3x^2+3x}{2x^2+2}$$ Is a good practice to do this? Change the $ -\infty $ to $ \infty $, and change the sign of the $ x $ variables: $$ \lim _{x \to \infty} ...
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4answers
130 views

Sequence of numbers with infinite number of primes

If I have an infinite sequence of positive integers with infinite number of primes and if I have an infinite number of distinct sequences with such properties may I claim that there is an infinite ...
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2answers
340 views

Proof of an infinite sum of probabilities [duplicate]

Possible Duplicate: Alternative Expected Value Proof If $X$ is a random variable that takes values in the range $\left \{ 1,2,3,4,5,6,\ldots \right \}$ how can I prove the following ...
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1answer
94 views

Divisibility with sums going to infinity

I can't quite wrap my head around this. Given the formula $(1-x)(1+x+x^2+...) = 1$ It seems clear to me why this is true. All the x terms cancel out and we are left with one. And this is clearly ...
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1answer
101 views

Solve an infinite sum

I need to find the sum of this series: $1, 2 \left ( 1 - \frac{1}{\sqrt{15}} \right ), 3 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 2, 4 \left ( 1 - \frac{1}{\sqrt{15}} \right ) ^ 3, 5 \left ( 1 - ...
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Does an infinite random sequence contain all finite sequences?

If we have a finite alphabet, with each letter having a non-zero probability of being selected, will an infinite sequence of letters selected from that alphabet contain all finite sequences of letters ...
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3answers
235 views

What is the definition of $\infty$

I saw in a note that say $\infty$ is not a real number, and there is no interval of the form $(a, \infty]$? So what is the definintion of $\infty$?
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1answer
217 views

Division by zero [duplicate]

Possible Duplicate: Division by $0$ Everyone knows that $(x/y)\times y = x$ So why does $(x/0)\times 0 \ne x$? According to Wolfram Alpha, it is 'indeterminate'. What does this mean? ...
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307 views

Doubly infinite numbers

Real numbers are sequences of integers which are infinite in one direction. If I have a string which is infinite in both directions, say ...345123985..., then I can form an injection from these ...
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1answer
408 views

Can one compare negative infinity and positive infinity?

I have always wondered, is negative infinity less than positive infinity? Can I compare them?
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3answers
442 views

What is the result of $\infty - 1$?

I was wondering after reading "What is the result of infinity minus infinity", is there any logical result for $\infty - 1$ ?
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3answers
981 views

Rejecting infinity

I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...
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2answers
324 views

One-to-One Mapping

I have a conceptual kind of a question in one of my lectures, where you have two line segments one is $(0,1)$ and the other $(0,2)$ which of course includes $1$ in the segment. It then ask what the ...
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3answers
224 views

How to define $-\infty$?

I think I understand the fundamental concept of infinity. Elementary mathematics define $\infty := \frac{x}{0}$, for every $x$. And also $\infty := \frac{-x}{0}$ for every $x$. I know only one ...
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4answers
341 views

The limit of binomial distributed random variable

Edit (As Robert pointed out, what I was trying to prove is incorrect. So now I ask the right question here, to avoid duplicate question) For infinite independent Bernoulli trials with probability ...
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3answers
151 views

Continuity of $f(x)$ involving infinity

$f(x)= \frac{\sin(\pi x)}{x(1-x)}$ How can I define $f(0)$ and $f(1)$ to make $f(x)$ continuous on $[0,1]$? I've found that the limit at $0 = \pi$, and the limit from the left at $1 = \infty$. I ...
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1answer
484 views

Is the Unicode designed assuming the Continuum Hypothesis?

The Unicode chart for "letterlike symbols" states that א 2135 ALEF SYMBOL = first transfinite cardinal (countable) ב 2136 BET SYMBOL = second transfinite cardinal (the continuum) I ...
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11answers
41k views

What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
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5answers
3k views

Sum of infinite divergent series

I have learned that positive infinity plus negative infinity isn't equal to zero, it's an indeterminate form. However what happens if we subtract two infinite divergent series ...
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3answers
461 views

Why $\frac{1}{\infty } \approx 0 $ and $ \frac{1}{0} = {\infty}$?

First I have checked the search option but found nothing relevant to my problem and also level of math. I just started learning the language of mathematics, on my own and I have trouble understanding ...
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1answer
286 views

Infinity - a simple question

This is a simple question, and maybe stupid: Is this true: $\infty < 1000\cdot\infty$ ?
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664 views

Is there an infinite number of primes constructed as in Euclid's proof?

In Euclid's proof that there are infinitely many primes, the number $p_1 p_2 ... p_n + 1$ is constructed and proved to be either a prime, or a product of primes greater than $p_n$. Trivially, we ...
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2answers
392 views

Comparing infinite numbers

Suppose you have 2 infinite numbers, say $A$ and $B$. $A$ is an element of the hyperreals, so that $A$ is greater than every real number. $B$ is the size of the set of natural numbers, $\aleph_0$ ...
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5answers
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Limit approaching infinity of sine function

I'd like to ask a question which I have been reflecting on for some time now. What is the limit of: $f(x) = \sin(x)$ as $x$ tends to infinity? As we know, the function has a definite value for each ...
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4answers
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The leap to infinite dimensions

Extending this question, page 447 of Gilbert Strang's Algebra book says What does it mean for a vector to have infinitely many components? There are two different answers, both good: 1) The ...
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3answers
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Math without infinity

Does math require a concept of infinity? For instance if I wanted to take the limit of $f(x)$ as $x \rightarrow \infty$, I could use the substitution $x=1/y$ and take the limit as $y\rightarrow 0^+$. ...
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5answers
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Negative 1 to the power of Infinity

Can anyone explain me what the result of $$\lim_{n\rightarrow\infty} (-1)^n$$ is and the reason?
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5answers
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Is infinity an odd or even number?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
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Why is $\infty^0$ indeterminate?

In a recent test question I was required to us L'Hopital's rule to evaluate: $$\lim_{x\to 0^+} x\ln{(e^{2x}-1)}$$ I assumed that anything multiplied by 0 would give an answer of 0. This turns out ...
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4answers
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Can a circle truly exist?

Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says... A circle is a simple shape of Euclidean geometry ...
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5answers
4k views

One divided by Infinity?

Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me. I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - ...
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4answers
633 views

What infinity is greater than the continuum? Show with an example

The diagonal argument establishes that the continuum is greater than countable infinity. What is an example of the next infinity (or any greater infinity) and how can it be shown that there is no 1:1 ...
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4answers
395 views

Golden ratio powers tend to integer values

If $G$ is the golden ratio, then $\lim_{n \to \infty}G^n$ tends ever nearer to integer values that approach $\infty$. Can it therefore be proved that $\infty$ is itself an integer? If not, why not?
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1answer
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Why does Cantor's diagonal argument not work for rational numbers?

If we map every integer to a string that represents a rational number, and produce a number different from all the ones listed, we are essentially following Cantor's algorithm. But why does it not ...
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3answers
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Regarding limits and $1^\infty$ [duplicate]

Possible Duplicate: Why is $1^{\infty}$ considered to be an indeterminate form I have some questions about limits and the undefinability of $1^\infty$. For example, is ...
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1answer
293 views

Notation for different sizes of infinity?

i realize that there are multiple sizes of infinity so one can be larger than another, but how do you show that one infinity is larger. I'm not looking for proofs or anything but I just want the ...
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7answers
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Infinity = -1 paradox

I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1: Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 ...
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9answers
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Is infinity a number?

Is infinity a number? Why or why not? Some commentary: I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school ...
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2answers
661 views

Infinite Prime Proof Using Euler's Totient

I need something explained or corrected: In my number theory class we proved that there are an infinite number of primes using Euler's Phi Totient. It went something like this: Let $M = p_1 p_2 ...
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3answers
424 views

Comparing infinite sets (of real numbers)

If $A$ is the set of all real numbers in $(0,1)$ with no $5$ in their decimal representation, and $B$ is the set with no $34$ and no $76446$. Then the set $B$ is in some sense larger then $A$, how can ...
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8answers
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Why is one “$\infty$” number enough for complex numbers?

Can anyone give me a rigorous explanation, why one needs only one number "$\infty$", when dealing with complex numbers, instead of 2 numbers $+\infty, \ -\infty$ like in the case, when dealing with ...
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1answer
146 views

Kappa function in infinite series

I saw a greek letter in an infinite series, and found out it was Kappa. What does this do? It looks like a giant K. http://www.wolframalpha.com/input/?i=find+continued+fraction+of+square+root That's ...
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3answers
207 views

Surface under $\frac{1}{x}$ is $\infty$, while surface under $\frac{1}{x^2}$ is $1$?

Since the antiderivative of $\frac{1}{x}$ is $\ln(|x|)$, the surface under the graph of $\frac{1}{x}$ with $x>1$ is $\infty$. However, the antiderivative of $\frac{1}{x^2}$ is $-\frac{1}{x}$, so ...
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3answers
489 views

How is it that this shape can converge to what looks like a triangle but has a different perimeter?

I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered ...
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10answers
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Why is Infinity multiplied by Zero not an easy Zero answer?

I did a bit of math at school and it seems like an easy one - what am I missing? $$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$ $$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ ...
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2answers
62 views

interval for a product to infinity

I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, ...
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1answer
2k views

Limit approaches infinity on one side and negative infinity on other side

I know this is a simple question for most of you, but I am currently studying for a Calculus exam and was just wondering why an online calculator I am using to double-check my work was disagreeing ...
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1answer
784 views

Justification for infinite KL Divergence

As I understand the KL Divergence, it measures how different two probability distributions $P$ and $Q$ are. However, say the two distributions are: ...
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3answers
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Are there any series whose convergence is unknown?

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will ...
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Does this expression represent the largest real number?

I'm not very good at this, so hopefully I'm not making a silly mistake here... Assuming that $\infty$ is larger than any real number, we can then assume that: $\dfrac{1}{\infty}$ is the smallest ...