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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-5
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8answers
132 views

All numbers are undefined? [on hold]

Notice the following set of equations $$\infty = \infty -x$$ $$x = \infty - \infty$$ $$x = \text{Not defined}$$ here $x$ can be any real number and hence all real numbers are undefined. This is ...
1
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3answers
41 views

Need assistance solving a limit question without applying l'hopital's rule

This is the question: $$\lim_{x \rightarrow-\infty} \frac{|2x+5|}{2x+5}$$ I know the answer is $-1$, but can someone go through the steps and explaining it to me?
0
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0answers
30 views

find the supremum

Hi I'm trying to figure out for which values of $w$ of $u(x,t)$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is ...
0
votes
1answer
35 views

If $A$, $B$, $C$ are any infinite sets then is $|A|=|B|$ and $|A|=|C|$ $\Longleftrightarrow |A|=|B\cup{}C|$?

Suppose we have three sets $A$, $B$, and $C$ that we know are infinite sets, but we do not know anything else about the cardinality of $A$, $B$, and $C$. Is $|A|=|B|$ and $|A|=|C|$ ...
1
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3answers
56 views

Uncountable infinity

The "number" of real numbers in $[0,1]$ is uncountably infinite, just as the "number" of real numbers in $[0,10]$ is uncountably infinite. However, my intuition would tell me the second interval has ...
0
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0answers
19 views

Compute a sum with finite and infinite elements

I would like to compute the following summation: $$ s = \sum_{i=1}^n a_i \, \Phi^{-1}(u_i) $$ where $\Phi^{-1}$ is the inverse of the standard Gaussian distribution function, $a_i$ are some real ...
1
vote
1answer
76 views

Question about infinity

That might be a silly question, but here goes: I see a lot of "big numbers" in physics, such as the size of the state space of all the particles in the visible Universe, and those numbers can be ...
1
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2answers
26 views

Infinite averages

If you wanted to find the average of infinite items, what would it be? Would an estimate of the first x averaged be a good estimate? Or would the value be nearly zero because you are dividing by ...
1
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0answers
33 views

How does one prove that two sequences are equal at infinity?

I came up with $e = \sum_{n=0}^\infty \frac 1 {n!}$ (see here) I am now trying to prove that this is equivalent to $\lim_{n\to \infty} {(1+\frac1 n)}^n$ In general, how would one go about such a ...
0
votes
1answer
38 views

Is the number of points on a plane larger than the number of points on a line?

The number of points on a line is uncountably infinite. The number of lines on a plane is uncountably infinite. It seems like it follows that there would be an uncountably infinite number of points on ...
0
votes
1answer
22 views

limits as $x\rightarrow\pm\infty$ of indeterminate forms $\frac{a^x+b^x}{c^x+d^x}$, where $a,b,c,d\in\mathbb{R}$

Good day sirs would you kindly help me to find the limit of $\frac{a^x+b^x}{c^x+d^x}$ as $x\rightarrow\pm\infty$, where $a$,$b$,$c$ and $d$ are real numbers? I already know how to use the L' ...
-1
votes
0answers
29 views

How many different random numbers can you generate between zero and infinity?

My theory is that you can't generate any random number between zero and infinity. But thats kind of unexpected because you can generate 100 different random numbers between 0 and 100 and 1000 ...
0
votes
2answers
70 views

Bijection between $\mathbb{Z} \longmapsto \mathbb{R}$ [duplicate]

I recently learned of Cantor's diagonal argument, and was thinking about why there can't be a bijection between any infinite set of integers and any infinite set of real numbers. I understood the ...
1
vote
2answers
73 views

Every II-finite set is III-finite

I need some help proving that if a set $X$ is II-finite then it is III-finite, i.e. if every non-empty family of subsets of $X$ which is linearly ordered by inclusion has a maximal element under ...
1
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1answer
24 views

Asymptotic behaviour of sequences

Could anyone explain in details how these approximations as $n \to \infty$ are found? ($a$ is a positive real number) ${x_n} = \frac{1}{n}\left( {\frac{a}{3} - \frac{3}{2}} \right) + O\left( ...
0
votes
1answer
34 views

Is the derivative of a exponential function a^x always greater than the derivative of a polynomial x^n as x approaches infinity

with n and a being any constants > than 1. I have tried taking the $\lim\limits_{x \to \infty} a^x / x^n$, and l'hopitals is telling me than $x^n$ can always be reduced to 1 with multiple iterations, ...
0
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1answer
46 views

Problems with x/∞ [duplicate]

If $\dfrac {x} {\infty }=0,$ where $x$ is a finite number, than wouldn't $0\cdot \infty $ be equal to any number? Making this not work?
1
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2answers
34 views

Complex infinity ($1/0$) [duplicate]

I've learned that $$1/0$$ is postive and negative infinity, but if I ask wolfram mathematica to calculate $$1/0$$ it gives me: 'complex infinity' but how can we proof that that is true?
1
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2answers
66 views

Prove or disprove : if $a_n$ has a limit and $b_n$ doesn't have a limit then $a_n + b_n$ doesn't have a limit

I think it's wrong but I couldn't find an example that disproves this. If this is true I need to prove it and if it's wrong I have to give an example to disprove it.
0
votes
2answers
36 views

P vs NP and Countable vs Uncountable Decision Space

I have noticed that whenever the scope of a problem is pushed to infinity, problems in NP have an uncountably infinite decision space whereas problems in P seem to have a countably infinite decision ...
0
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1answer
43 views

Number of paths in a graph with infinite nodes

Does a graph with infinite nodes that is not fully connected have a countably infinite or a uncountably infinite number of paths originating from a single node? We are only concerned with paths that ...
1
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1answer
31 views

Doubt related to the extended real line and distance/metric

I am studying real analysis and I am being introduced to functions that take values on the extended real line, I have a fundamental doubt about this so I'll give an example to illustrate my confusion: ...
-6
votes
4answers
102 views

Proof of 1+1+1+1+… = 0 [closed]

I have thought that 1+1+1+1+... is equal to -1/2. However I found this proof based on the assumption of 1+2+3+4+5+... = -1/12 and -1+(-2)+(-3)+(-4)(-5)+... = +1/12 1+1+1+1+1+1+... is equal to ...
0
votes
3answers
111 views

All sets of rational numbers are bigger than the set containing infinite integers - or are they?

Intro This started with me learning the different types of infinity. I like to call them types instead of sizes due to the fact, that infinite is defined by being endless or "not-finite" - meaning ...
0
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1answer
86 views

Extend ${\bigl(1+\frac1x\bigr)}^{{x}}$ to $\overline{\mathbb R}$

We can extend these functions to $\overline{\mathbb R}$ by taking limits says here. \begin{align} \mathrm e^{-\infty} &= 0 \\ \mathrm e^{+\infty} &= \infty \\ \ln{\left|0\right|} &= ...
1
vote
1answer
17 views

Limit at $\infty$ of a polynomial multiplied by a negative exponential

I am trying to show $\int_0^{\infty} x^2 e^{-2 x} dx = 1/4 $ Integration by parts gets the indefinite integral $$\int x^2 e^{-2 x} dx = \frac{-1}{4} e^{-2 x} (2 x^2+2 x+1)+constant$$ In order to ...
4
votes
3answers
422 views

infinite monkey problem - probability of an infinite sequence containing an infinite sequence [duplicate]

Note: This question is specifically about when the infinite monkey theorem is extended to reproducing an infinite sequence (as oppose to a finite one) I was browsing wikipedia, and came across the ...
0
votes
2answers
31 views

Showing that $\log(\log(n))^{\log(n)}$ is $O(7^{\sqrt n})$

What's a straightforward way to prove that $\log(\log(n))^{\log(n)}$ is $O(7^{\sqrt n})$? (I'm dealing with Big O Notation)
2
votes
4answers
79 views

What is limit of $\lim_{x\to\infty}((\frac{a^x+b^x}{2})^{1/x})$?

How I can calculate limite of this equation?! It can be solved using a famous theorem but I forgot it, may someone help me to calculate and prove it or even remind me the theorem? $$ ...
0
votes
1answer
97 views

What's the largest number

Originally this question started as 'what is the largest number' using $\aleph_0$ as a start, and continuing using concepts such as ${\aleph_0}^{\aleph_0}$, and Knuth's Tower notation $\uparrow$, so ...
1
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3answers
114 views

Is it correct to say that $\lim_{x\to\infty}e^x=\infty$?

I saw $$\lim_{x\to\infty}e^x=\infty$$ in a textbook, but I think the limit of the left part doesn't exist. So left part doesn't equal right part. Am I right?
2
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6answers
87 views

Proof of sum results

I was going through some of my notes when I found both these sums with their results $$ x^0+x^1+x^2+x^3+... = \frac{1}{1-x}, |x|<1 $$ $$ 0+1+2x+3x^2+4x^3+... = \frac{1}{(1-x)^2} $$ I tried but I ...
1
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2answers
43 views

Maclaurin series not giving right answers when manually deriving?

Apologies about any formatting issues, I am new. I have to find the first four terms of the Maclaurin series for $$f(x) = \frac{1}{1-x}$$ So first I plug in: 1st term is 1 Then derive $$(1-x)^{-1} ...
1
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3answers
83 views

Convergence of an infinite series involving conjugates

I have the infinite series $$\sum_{n=1}^\infty \left(1-\cos\frac{1}{n}\right) $$ I have to find if it converges or not, and I know I have to use the conjugate find it. So I get ...
1
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2answers
57 views

Integral of $\frac{\sin x}{1+\sin^2x}$ from 0 to $\pi/2$

I am trying yo find $\int_0^{\pi/2}\frac{\sin x}{1+\sin^2x}dx$. So far I have tried using the substitution $\tan u=\sin x$ which led me to $$\int_{u=0}^{u=\pi/4}\frac{\sin x}{\cos x}du$$ ...
4
votes
3answers
102 views

Can I deduce ZFC Standard from “ZFC Dedekind”?

ZFC Standard: Infinity, Extensionality, Specification, Pairing, Union, Replacement, Power Set, Choice and Regularity. ZFC Dedekind: Infinity replaced bij Dedekind Inifinity, other 8 axioms the same as ...
0
votes
1answer
50 views

Infinity figure of eight

I'm trying to build some jewellery in a 3D cad package. I found this: The function that draws a figure eight But I don't understand the equation (I only got to A' level :) Is there a way of ...
1
vote
1answer
52 views

Integral of $\frac{4}{x+1}$ from $4$ to $\infty$.

I want to calculate the integral $$\int_4^\infty\frac{4}{x+1}dx.$$ I know that the result is $$\lim_{x\to\infty}(4 \ln (x + 1)- 4 \ln (5)),$$ then I get $\infty - \ln (625)$. Is it still ...
27
votes
8answers
3k views

Sum of all integers

No, I'm not talking about $-\frac{1}{12}$. I was talking with someone the other day, and they said that the sum of all integers, positive and negative, is zero because they all cancel each other out. ...
0
votes
1answer
33 views

limt of the function as $\mu\rightarrow\infty$ or $\mu\rightarrow-\infty$ .

$\lim_{\mu\rightarrow\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ Also, $\lim_{\mu\rightarrow-\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ I know, ...
0
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1answer
35 views

Negative infinity to square equals positive infinity?

Is $$ -\infty^2 $$ always positive just like for ex. $$ (-2)^2 $$ is always positive?
3
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3answers
74 views

proving that the set of all english words is countble. [duplicate]

This is the question : Prove that the set of all the words in the English language is countble (the set's cardinality is אo) A word is defined as a finite sequence of letters in the English language. ...
0
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0answers
38 views

Limit of a sequence question

I have the following question in my assignment which I couldn't solve: Let $({a_n})$ and $({b_n})$ be two sequences, such that $\lim\limits_{n \to \infty}({a_n}{b_n}) =0 $ I have to prove if the ...
0
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1answer
52 views

Limit of $\frac {1}{x^2(x+7)}$ as x approaches $-7^-$

$\displaystyle \lim_{ x \rightarrow -7^{-}} \; \frac{1}{x^{2}(x+7)}$ My manual computation yields a different answer to that of Wolfram's. And I don't understand, why isn't the correct answer: the ...
0
votes
1answer
64 views

Given that there are infinities of different sizes, what is the biggest infinity? [closed]

Does it even make sense to talk about the biggest infinity? I have already seen this similar question's responses on the numbers of infinities (which I note has no accepted answer): How many ...
0
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4answers
35 views

Limit Equals Infinity for a Sequence

I have to solve the following question in my assignment: Prove that: $\lim\limits_{n \to \infty} \frac{n^2-n}{n+2}$ = $\infty$. I have to prove this with the following definition: "A series ...
0
votes
1answer
355 views

why does e raised to the power of negative infinity equal 0?

Why is it that e raised to the power of negative infinity would equal 0 instead of negative infinity? I am working on problems with regards to limits of integration, specifically improper integrals ...
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2answers
62 views

a question of infinity [closed]

If infinity or infinities cannot be physically proved ie actually counted, how do we know they really exist
14
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6answers
1k views

Can $f(\infty)$ be defined if the sequence $f(n)$ is divergent?

Let there be given an real-valued function $f(n)$ , with $n\in\mathbb{N}$ , $a,b \in\mathbb{R}$: $$ f(n+1) = a f(n) + b $$ What then is the value of $f(\infty)$ ? As a physicist by education, being ...