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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Why can't consecutive irrational numbers be treated mathematically as limits?

I'm a relative newcomer to these stackexchange websites, and this post will serve as my introduction to the Mathematics stackexchange site. After perusing some of the related questions, I found these ...
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12 views

integration featuring the unit step function

Compute the following integrals I don't know how to use MathJaX so here's a link to the image of the integrals where u(t) is the unit step function and σ is some variable of integration
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1answer
57 views

On Proving that the first euclidean axiom is wrong [on hold]

Well, The first axiom in the euclidean geometry is "A straight line segment can be drawn joining any two points". But I think that there are points that can't be joined: In the image below, We have ...
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1answer
29 views

Arithemetic series addition

Lets say I have M= 1+2+3+4+5+6+7.... (to infinity) and I have another sequence,N= 6+14+22+30..... (to infinity) is it possible to say that N = 4M +2 ? Or is there another way that I can write ...
5
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3answers
125 views

Proof of Nesbitt's Inequality?

I just thought of this proof but I can't seem to get it to work. Let $a,b,c>0$, prove that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$$ Proof: Since the inequality is homogeneous, ...
0
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0answers
35 views

Evaluating the antiderivative of a particular improper integral

The task is to integrate $$\tau = \int\limits_{-\infty}^{+\infty}\frac{dx}{\sqrt{E - \frac{U}{cosh^2(ax)}}}, E>U$$ but after taking the integral I get $$\tau ...
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1answer
33 views

Help with calculus 12 question [on hold]

So I am having trouble with my calculus homework! The question is: Find a value for $a$ so that the function $$f(x)=\begin{cases}x^2-1& x<3\\2ax& x\ge 3\end{cases}$$ is continuos. ...
3
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1answer
44 views

Why do we care about the 'rapidness' for convergence?

It is those puzzeling improper integrals that I can't get my head around.... Does the (improper) integral $\frac 1{x^2}$ from 1 to $\infty$ coverges because it is converging "fast" or because it has ...
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0answers
50 views

Quantifying infinitely large sums such as $\sum_{x\in\mathbb{R}^+} x$

I thought of this as a student in calculus years ago, and it may be a silly kind of question. I wondered if there were notions of different sizes of infinity a series might sum to, which then lead me ...
0
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1answer
39 views

Recurrence relation taken to infinity [closed]

Very simple question.... Say we have a recursive function: $x_n = \dfrac{c}{2ax_{n-1}}$ where $x_0 \in \mathbb Z^+$ and $n \in \mathbb Z^+$ ($x_n \in \mathbb Z^+$, for now, but this should work ...
0
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2answers
41 views

infinite limit question from Calc I

Find the limit $$\lim_{x\to\infty}\sqrt{x^2+x+1}-x$$ This limit is part of a question involving squeeze theorum, the limit is $\frac12$ but i don't know how to prove it because of the polynomial in ...
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1answer
107 views

How is infinity not a number? [closed]

I've heard this surprising thing somewhere and it seems to me that infinity isn't a number, but how is that true? I want to know by the answers you put below (a rhyme! (know and below rhyme))! ...
5
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5answers
384 views

L'Hôpital's as $x$ tends to infinity

I'm searching for the explanation to the limit of: $$ \lim\limits_{x\to\infty} x\, \ln\frac{x+1}{x-1}. $$ I know the answer is 2, but I can't seem to get there. The problem is in my textbook under a ...
0
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2answers
17 views

Convergence of this alternating series?

I "heard" the following formula for any $C \ge 1$: $\sum\limits_{k=0}^\infty \dfrac{(-1)^k}{(k+1)C^k} = C \log \dfrac{C+1}{C}$ Is it correct? What would be a proof?
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0answers
8 views

Estimating the mean Euclidean distance between two overlapping, not-matching shapes

I’d like to determine the mean distance between two irregular overlapped, not-matching shapes ($X$ and $Y$). In $Figure 1$, $X$ is “visually above” $Y$, and that’s why we can’t see part of the $Y$ ...
3
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1answer
130 views

Did I construct an infinite set equal to $\{1\}$?

Okay, I'm trying to understand the argument that NJ Wildberger gives in the following video: https://www.youtube.com/watch?v=5CiiGdaYEPU He tries to explain why he things infinite sets don't make ...
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1answer
26 views

Question about $\lim_{x\to \infty}\frac{\cos(3x)}{e^{8x}}$

$\lim_{x\to \infty}\dfrac{\cos(3x)}{e^{8x}}$ The answer is $0$. Why is the answer $0$? The top oscillates between $-1$ and $1$ and the bottom becomes huge, but since the top is oscillating, ...
2
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4answers
56 views

The limit as x approaches infinity

$$\lim_{x\to\infty}x\left(1-\sqrt{1+\frac1{2x}}\right)$$ Can anyone explain how to get this?
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5answers
384 views

Limit to Infinity question?

$$\lim_{x\to\infty}\left(-\sqrt{-2x+x^2}+\sqrt{2x+x^2}\right)=2$$ I'm not sure how to go about solving this problem.
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2answers
34 views

Partial sum formula of a polynomial series?

I am trying to find the partial sum formula of the following series: $$ \sum_{y=1}^{\infty} \frac{4y^2-12y+9}{(y+3)(y+2)(y+1)y} $$ I have tried using Faulhaber's formula without success. I have also ...
0
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1answer
57 views

Is$\ \infty \times 0$ undefined in the extended real numbers?

And if it is, why? Is it a kind of postulate related to the fact that infinitely many points make a line?
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3answers
38 views

How can I evaluate this limit?

I'm studying for an upcoming midterm and i'm stuck on this question. It's asking me to evaluate the following limit and justify my answer. $\lim \limits_{x \to \infty} \sqrt{x^2+3x} - \sqrt{x^2-2x}$ ...
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1answer
27 views

What is the geometric way of relating zero to infinity?

I once saw (what I think was) a geometric way a relating zero to infinity. Something about a circle with radius 1 around the origin. Can you tell me where to find that? Thanks
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2answers
53 views

When is an infinite set larger than another infinite set?

Somewhat of a basic question that I've been pondering about, suppose we have 2 finite sets $A,B$, arbitrary sets with arbitrary elements that we know nothing about, except that they are both finite. ...
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3answers
39 views

The limit of $\sqrt{x^2+x+1}-\sqrt{x^2+1}$ as $x\to\infty$ [closed]

Currently I'm self studying limits. but I don't know how to get the answer to this question: $$\lim _ { x\to \infty }\left(\sqrt{x^2+x+1}-\sqrt{x^2+1}\right)$$ can someone help me
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1answer
29 views

What is the cardinality of all frames in time?

If we divide time into individual frames, then we would get a set of infinite frames. But what is the cardinality of such a set? Since time is continuous, like the real numbers, I would expect the ...
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0answers
40 views

Hilbert's hotel with uncountably infinite rooms: can you fit $\mathbb R^2$ guests?

I'm trying to expand on Hilbert's paradox. The original version states that: Suppose there is a hotel with a countable infinity of rooms (eg. $\mathbb N$), all of which are occupied. ...
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2answers
58 views

Limit that fail to exist

Does a limit that equals to infinity considered to exist ?? am confused !! for Example 1/(x-2)--> when evaluating the limit at 2 the result is 1/0 which is infinity while after looking at the graph ...
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0answers
34 views

Is$\ +\infty$ greater than any other number (surreal, superreal, hyperreal, …)?

Let$\ \mathbb{A}$ be an arbitrary totally ordered set and consider the largest element of the set of extended real numbers,$\ +\infty$. Can we say that$\ +\infty > \chi $, for *any*$\ \chi \in ...
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2answers
40 views

Question about $\lim_{x \to -\infty}\frac{\sqrt{10+11x^2}}{12+13x}$

$\lim_{x \to -\infty}\dfrac{\sqrt{10+11x^2}}{12+13x}$ = multiply top and bottom by $\dfrac{1}{x}=-\dfrac{1}{\sqrt{x^2}}$ My question is, why is the negative sign in front so crucial, I don't ...
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1answer
27 views

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$. My attempt: Let $x_n\to -\infty$ and $\epsilon\gt 0$. By the Archimedean Principle pick $N\in \mathbb N$ such that ...
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3answers
56 views

How do I solve $\lim_{x\to -\infty}(\sqrt{x^2 + x + 1} + x)$?

I'm having trouble finding this limit: $$\lim_{x\to -\infty}(\sqrt{x^2 + x + 1} + x)$$ I tried multiplying by the conjugate: $$\lim_{x\to -\infty}(\frac{\sqrt{x^2 + x + 1} + x}{1} \times ...
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63 views

Does this sequence diverge to ∞?

The sequence $(a_n)_{n \geq 1}$ is defined as follows: $$a_n:= \begin{cases} 0 \quad \text{if} \quad n \quad \text{is odd}\\ n \quad \text{if} \quad n \quad \text{is even}\end{cases} \quad .$$ Does ...
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7answers
2k views

There is no smallest infinity in calculus?

Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess. From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and ...
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3answers
58 views

Calculate exact value of and infinite sum [duplicate]

Im trying to find the exact value of the infinite sum : 3 + 1/3 + 1/27 + 1/243 + 1/2187 + ... I can see that to generate new terms we take the previous term and divide by 9 or multiply by 9. Not ...
2
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0answers
46 views

Is this set countably infinite or not?

"Far away, in the heavenly abode of the great god Indra, there is a wonderful net that has been hung by some artificer in such a manner that it stretches out infinitely in all directions. In ...
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40 views

Circles and the continuum hypothesis

I was trying to understand the undecidable nature of the continuum hypothesis and came up with the following question: The set of circles with a rational diameter is countably infinite (with ...
3
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3answers
120 views

Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...
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2answers
45 views

Limit approaching to negative infinity.

Q. Find $\lim _{x\to -\infty }\left(\frac{x^4\sin\frac{1}{x}+x^2}{1+|x|^3}\right)$ By inserting $x=-\frac{1}{y}$ and as $_{x\to \:-\infty \:}$ then $_{y\to \:0\:\:}$. By applying this my text arrive ...
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1answer
23 views

Help explain the set being constructed in this Cantor-Schroder-Berstein proof

The Cantor-Schroder-Bernstein theorem states that: Suppose $A$ and $B$ are sets. If $|A|\le |B|$ and $|B|\le |A|$, then $|A|=|B|$ Proof: So, $|A|\le|B|$ implies we can choose an injection ...
3
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1answer
44 views

Does there exist a connected 2-regular uncountable graph, or an uncountable path?

Does there exist a connected 2-regular uncountable graph? Can I use the axiom of choice to construct an uncountable path of elements from the reals? The question arose when reading this: Also, ...
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Sizes of infinity

I was just thinking about infinity (as you do) and thought the following. "There are infinitely many reals in the interval $x\in[0,1]$ and an 'equal number of reals' $x\in[1,2]$, so there are 'double ...
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2answers
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Is there a thing named a “spiral plane” which is a plane but it's spiral?

Hello, I'm wondering if there is such thing like this. Is there a plane which is not flat but spiral and extending for infinity? I have drawn a representation for what I mean but it's not thorough ...
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7answers
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Is 0.9 repeating = 1 disproved by asymptotes?

I'm discussing proofs that 0.9 repeating equals 1 with some friends, and they use asymptotes to disprove this. One says if we had the function $y=x/0.000\ldots1$ (and he's only using that impossible ...
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1answer
33 views

Degrees of freedom in each domain in Discrete, Continuous and Mixed Fourier Transforms

I'm having trouble with the different infinities involved in the Discrete and Continuous Fourier Transforms. In the DFT, we have a finite number $N$ time domain samples $x(i), 0\leq i<N$, which ...
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1answer
27 views

Help to prove this inductively defined function is surjective

Suppose that $A$ is a infinite subset of $\mathbb{Z}^+$. We construct a bijection $f:\mathbb{Z}^+ \rightarrow A$ and define $f(n)$ inductively as follows: Base case: Let $f(1)$ be the least element ...
2
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4answers
262 views

Can't find limit tending to infinity of a sequence

I'm stumped by $$\lim_{x \to \infty}\frac{1+3+5+\cdots+(2x-1)}{x+3} - x$$ My obvious first step was to get a lowest common denominator by $x(\frac{x+3}{x+3})$, giving $$\lim_{x \to ...
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Check proof of union of denumerable sets is denumerable too

I need to prove: If $A$ and $B$ are denumerable sets then so is their union $A\cup B$. In this case, denumerable is defined as: A set $X$ is said to be denumerable if there is a bijection ...
2
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1answer
65 views

The meaning of infinite series $\sum_{i=0}^\infty 2^{-i}$, its relation to partial sums and Cantor's diagonal argument

Let's define $S(n)$ as $S(n) = \sum_{i=0}^n 2^{-i}$. Obviously, $\lim_{n \to \infty} S(n) = 2$ and also $\forall n \in \mathbb{N}, S(n)<2$. Now my questions are about $Q = \sum_{i=0}^\infty ...
2
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1answer
79 views

Are $+\infty$ & $-\infty $ elements of the real number line? [duplicate]

Can someone give an explanation/proof of whether these two numbers lie on the real number line?