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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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 ...
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1answer
23 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( ...
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1answer
33 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, ...
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1answer
44 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?
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2answers
33 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?
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2answers
55 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.
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2answers
30 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 ...
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1answer
42 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 ...
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0answers
31 views

Something to the power of infinity is equal to? [on hold]

What is a number, when raised to the power infinity, equal to ($n^\infty$)? What are all its cases? Is it that an infinite power can be only carried out when limit is present?
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1answer
29 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: ...
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4answers
94 views

Proof of 1+1+1+1+… = 0

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 ...
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3answers
107 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 ...
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1answer
84 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|} &= ...
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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 ...
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3answers
415 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 ...
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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)
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0answers
66 views

treat Infinite as a quantity to resolve Zeno's dichotomy paradox [closed]

You can view my article here(hosted on google drive). I think I find an intuitive way to resolve Zeno's dichotomy paradox, provided we define infinite as a quantity, I know this definition would be ...
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4answers
78 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? $$ ...
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1answer
96 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 ...
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3answers
112 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?
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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 ...
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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} ...
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3answers
81 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 ...
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2answers
55 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 $$\int_{u=0}^{u=\pi/4}\frac{\sin x}{\cos x}du$$ ...
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3answers
100 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 ...
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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 ...
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1answer
51 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 ...
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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. ...
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1answer
32 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, ...
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1answer
33 views

Negative infinity to square equals positive infinity?

Is $$ -\infty^2 $$ always positive just like for ex. $$ (-2)^2 $$ is always positive?
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3answers
70 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. ...
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0answers
36 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 ...
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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 ...
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1answer
63 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 ...
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4answers
34 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 ...
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1answer
202 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
61 views

a question of infinity [closed]

If infinity or infinities cannot be physically proved ie actually counted, how do we know they really exist
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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 ...
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1answer
16 views

Series having strict inequality implies limits having strict equality?

I was wondering if I have two convergent series, say, $\sum_{n=1}^{\infty} s_n = s$ and $\sum_{n=1}^{\infty} t_n$ = t, and for all their partial sums we have that: $s_n > t_n$. Is it then ...
4
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1answer
79 views

Apples and Infinity

I am taking a proof writing and discrete mathematics course and we are learning about infinity. My TA asked me the following question and I'm wondering if my solution is correct? Question:Suppose ...
36
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3answers
2k views

Why can you chose how to align infinitely long equations when adding them?

I saw in a video this proof: Take this equation: $$f=1+\frac12+\frac14+\cdots$$ and do this: $$\begin{align} f&=1+1/2+1/4+\cdots\\ -\quad f/2&=\quad\:\:\:1/2+1/4+\cdots\\ ...
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2answers
58 views

what is the integral on $[0,2]$ of $x/(3-2x)$

i know that this is an improper integral, but when you evaluate the limits as $x\to (3/2)^-$ and $x\to (3/2)^+$, you get positive and negative infinity but I am not sure if you can cancel them ...
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2answers
45 views

Help with an improper integral!

I need some help with an indefinite integral problem (only the $2^{\textrm{nd}}$ part thou). Problem is as follows. Consider the function $f(x) = \dfrac{\ln\!\left(x\right)}{x^{p}}$, where $p>1$ ...
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1answer
66 views

Geometric definitions of infinity

There are several definitions of "infinite set" that are common in set theory. For instance, a set $S$ is finite if there is a bijection between a natural number $n$ and $S$; it is infinite ...
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3answers
376 views

How to prove the infinite number of sides in a circle?

I was in geometry class today when I came across the following formula for the external angle of a regular polygon with n sides: $$Ea = \frac{360º}{n}$$ So I thought if $$ n\rightarrow\infty $$ then ...
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0answers
20 views

Mapping from 'one infinity to another'.

I've been wondering about this for a while now. Consider the function $f(x)=\frac{1}{x}$. When $x\in(0,1)$, the function maps to the interval $(1,\infty)$. Conversely, anything in the interval ...
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1answer
94 views

Infinity equals zero

I am not a math whiz by any degree, so go easy on me if I've made a huge oversight. I am working on a geometrical proof for a philosophy project I am working on and I need some help with its current ...
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1answer
28 views

limit of a floor function

I came across the following limit in a math book $\lim_{x\to \infty}\frac{(x^x)}{E(x)^{E(x)}} $ where $E(x)$ represent the floor function, and the question was to prove that this limit doesn't ...
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59 views

If an object halves its speed every second (but never gets to 0), will it eventually get from point A to point B?

There is a ball that starts at point A on a line and moves toward point B. Every second, it moves half of the distance left, but never stops moving: Etc. Would the ball ever reach point B? In one ...