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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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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439 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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32 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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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
100 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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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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93 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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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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91 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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58 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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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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54 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
55 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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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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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, ...
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43 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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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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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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4answers
43 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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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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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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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 ...
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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 ...
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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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59 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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48 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
81 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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525 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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28 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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124 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
35 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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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 ...
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41 views

infinity sum of the fractional

Can anyone explain how to simplify $ \frac{2}{3} + \frac{6}{9} + \frac{12}{27} + \frac{20}{81} + \frac{30}{243} + . . . $ I have no any idea since i dont have pattern i can't do it with integral or ...
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How do I show that infinite application of this function gives a constant?

I want to show that g(x) returns the same value independent of x and hence is a constant. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
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56 views

Find the equation of a line parallel to the y-axis, that goes through the point $(\pi,0)$

I have been trying to do this problem and I am very confused. I know the gradient is infinity when any line is parallel to the y-axis, therefore, $y = \infty \cdot x + c$, right ($y = mx + c$ being ...
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2answers
55 views

Reversing a number of infinitely many digits

Lets say we have a function that gets as input a real number and returns its reverse e.g. 123.12 -> 21.321 So what happens when the input is a number α that has infinitely many digits. Does then ...
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100 views

Why doesn't this proof work? (Zero times infinity equals zero)

Say we are trying to prove $$ 0\cdot n = 0 $$ By mathematical induction, we start with a base case of n = 1 $$ 0\cdot 1 = 0 $$ So now we assume our original formula is true, and try to prove a case ...
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64 views

Is the integral of the sum really the sum of the integrals?

I was asked to find the mclaurin series of $\int_0^x\frac{\arctan (t)}{t}dt$ using the known mclaurin for arctan: $\arctan(t)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}t^{2n-1}}{2n-1}$ Ok, so what I did ...
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1answer
106 views

Does an unbounded straight line have infinitely many axes of symmetry?

In a circle, any diameter is an axis of symmetry, so technically a cirle should have infinitely many axes of symmetry. This got me thinking about the axes of symmetry of straight lines. A line ...
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23 views

Convention on infinity comparision

What's the difference between $-\infty \leq a \leq \infty$ and $-\infty < a < \infty$ conceptually or otherwise. It doesn't really affect the solution to the solution of this problem I'm ...
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27 views

Calculating limits using the definition of number e

I have some examples in Demidovič using this technique and there seems to be no reliable source for them online, so I'll make a small tutorial. Example 1: ...
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101 views

DTFT of Impulse train is equal to 0 through my equation.

Let me have an impulse train function as below, $$ x[n] = \sum_{m=-\infty}^{\infty} {\delta[n-f_0 m]} $$ where, $f_0 \in \textbf{Z}$. Now, I am trying to calculate its DTFT, so I put it into DTFT ...
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What is the meaning of “uncountably infinite” within countable models of set theory?

So, I don't know much about countable models of set theory, other than that they exist. To me, their existence is a very weird thing (and a reason to move away from first-order formulations). Here is ...
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Filpping a coin infinitely many times

If a coin was flipped an infinite number of times, is it guaranteed to be heads at least once?
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Real analysis - Countable and uncountable set

I'm having a problem understanding this: The union of a countable set and an uncountable set is uncountable. Help me please!
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121 views

The sum of $1+1+1+1+…$

My teacher recently showed me a rather weird result and I would like to know if he was just tricking me or if he was serious. He showed me that $g=1-1+1-1+1-...=\frac{1}{2}$ Then he said that ...
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Suppose some theory T has countably many axioms, how many models of $T$ are there of cardinality $\aleph_1$,$\aleph_2$,$\aleph_{\omega_1}$?

Setting Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes. So we see $T$ has ...
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53 views

Evaluating $\lim_{x\to -\infty} \frac{(x-1)}{(x^{2/3}-1)}$

The limit at negative infinity should not exist, right? $$\lim_{x\to -\infty} \frac{(x-1)}{(x^{2/3}-1)}$$ for positive infinity, the limit is infinity, but the function is undefined for values less ...