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 sum of the infinite series [duplicate]

The sum of the infinite series: $$ \frac{1}{2} +\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +\cdots$$ I am able to find ...
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I need advice on calculating this limit of a function resulting in $-\frac14\pi$

I am looking for advice on solving this limit of a function. I am struggling to find the correct process: $$\lim\limits_{x\to-\infty}\operatorname{arccotg}\frac{x}{(x^2-4)^{\frac12}}$$
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Are the integers closed under addition… really?

Okay so I'm a 3rd year undergraduate studying Mathematics. I've proved in group theory countless times that the integers are closed under addition. It's obvious to me that they are. However this has ...
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Non-standard numbers and exponential form of Zeta function

Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...
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1answer
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What is ordinal expression of $\infty$? [closed]

$\infty$ - is cardinal expression? ?Origin of a line ray is ordinal expression of $\infty$, if distance from the $\infty$ to the $0$ origin of line ray?
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What is the meaning of Right Hand Limit at $\infty$?

For a limit to exist, the left hand limit must equal the right hand limit. That is, $$\lim_{x\to c^+}f(x)=\lim_{x\to c^-} f(x)$$ However, if $x\to\infty$, then what does the right hand limit mean ? ...
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dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

I have a question about this integral $ \displaystyle \int_{-\infty}^{+\infty} \delta'(x-3)e^{x^2}dx $ by integration by parts I get; $ \displaystyle ...
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4answers
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Why Zero divided by Zero is undefined and not Infinity [duplicate]

apologize in advance if this is a duplicate, but I found a lot questions related to this but none answering this specific question. My logic is: let's consider division the opposite of ...
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What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
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Good papers on Cantor's infinities?

I'm trying to get a mathematical treatment on Cantor's infinities, but can't seem to find papers merely by googling. Any recommendations?
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55 views

An infinite dictionary: countably infinite or uncountably infinite?

This question concerns Ian Stewart's "Hyperwebster", an uncountable dictionary. Say a publishing company wants to publish every possible permutation (of any length) of the characters A-Z. The ...
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Why is there antagonism towards extended real numbers?

In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept ...
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30 views

Comparing Infinities

Is it ever possible to say that $\infty = \infty$? For example, does the number of odd numbers ($\infty$) equal the number of even numbers ($\infty$)? Does does the number of odd numbers ($\infty$) ...
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Whats infinity divided by infinity?

This should be a simple question but i just want to make sure. I know from infinity/infinity is undefined. However if we have 2 equal infinities divided by each other it would be 1? And if we have ...
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2answers
25 views

Proving existence of numbers with intermediate value theorem

How do you use the intermediate value theorem to prove the existence of numbers? For example, with $f(x) = c^2 = 2$, how can I prove that $\sqrt2$ or a positive number $"c"$ such that $f(x)$ is true ...
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Can a sequence have infinitely many limits among its subsequences?

Suppose we have an extended real (countably) infinite sequence $(x_n)$. Then consider all of its possible subsequences $(x_{n_k})$. We could then consider the set $$A = \{a\in ...
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1answer
52 views

Coupon collector problem doubts

The Coupon Collector problem off Wikipedia: Suppose that there is an urn of $n$ different coupons, from which coupons are being collected, equally likely, with replacement. How many coupons do you ...
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1answer
19 views

The Limit of an Integral Containing Exponentials

I am unsure how to show this. Suppose $\delta(s)$ defined on $(-\infty , s_*)$ is increasing and satisfies $\lim _{s\rightarrow s_*} \delta = \lim _{s\rightarrow s_*} \frac{d \delta}{d s} = \infty$ ...
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Is $\frac00=\infty$? And what is $\frac10$? Are they same? Does it hold true for any constant $a$ in $\frac{a}0$ [duplicate]

I know that $\lim_{x\to0}\frac{x}{x}=$ 1. But in my text book, it is written that it is $\infty$ and even $\frac10=\infty$. But how is it possible? And are they both same? What is the difference ...
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Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? [closed]

Here is a small intuition why it should be the later. Let $\omega$ be the number of all natural numbers. Then what is the smallest real number? We can write reals in binary form. Usual logic would ...
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1answer
26 views

Is there a relationship between the unbounded infinities and uncountable infinities? [duplicate]

When a function f increases without bound we say $f(x)=\infty$. How does this idea relate to, if at all with the infinite sets we study in set theory? To give a better understanding of why I'm ...
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Double Summation Trick

I have seen a couple of times the trick where $\displaystyle\sum_{i=1}^\infty \sum_{j=i}^\infty f(j)$ becomes $\displaystyle\sum_{j=1}^\infty \sum_{i=1}^j f(j)$ How does this work? I am so confused. ...
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Logic question requiring axiom of choice

Predicting Real Numbers Regarding the above question, the solutions require creating classes of sequences with representative sequences. How are those sequences constructed? How is it possible to ...
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722 views

Why do we distinguish between infinite cardinalities but not between infinite values?

More specifically, why are we "allowed" to denote $|\mathbb{N}|<|\mathbb{R}|$ but not $\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$? Can we distinguish between "countable ...
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1answer
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Existence of infinite set and axiom schema of replacement imply axiom of infinity

I'm self-teaching an intro to set theory course, and came across this exercise: Show that the existence of an infinite set is equivalent to the existence of an inductive set. For the notion of ...
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Alternate Axiom of Infinity

The Axiom of Infinity states that there is a set $S$ containing $\varnothing$ such that if $x$ is an element of $S$ then so is $x\cup\{x\}$. Is the following variant equivalent? There exists a ...
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Two similar method to calculate one equation get different answer

Method1:$$\lim_{x\rightarrow0}({\frac{e^x+xe^x}{e^x-1}}-\frac1x)=\lim_{x\rightarrow0}({\frac{e^x+xe^x}{x}}-\frac1x)=\lim_{x\rightarrow0}(\frac{e^x+xe^x-1}{x})=\lim_{x\rightarrow0}(2e^x+xe^x)=2$$ ...
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51 views

Is there a change-of-variables solution for integrals from negative infinity to a constant?

I found a fantastic and generalizable substitution technique for computing definite integrals that go to infinity from either negative infinity or a constant, regardless of the function (sorry for the ...
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2answers
177 views

Bijection between an infinite set and its union of a countably infinite set

I have $A$ as an infinite set and $S$ as a countably infinite set, (so that means there exists a one-to-one correspondence between $S$ and $\mathbb{N}$). How do I show that there always exists a ...
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2answers
48 views

What is the value of $0\times \infty$? (in $[0, +\infty]$)

In $[0,+\infty)$, $0^+\times +\infty$ can be any number in $(0,+\infty)$ so is undetermined; (in which $0^+$ means when a variable approaches to $0$). Because $\lim_{x\rightarrow ...
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Is 1 + 1 + 1 … a finite number? [closed]

Just a simple question whether the "number" $x = 1 + 1 + 1 ...$ is a finite number. On one hand, you can think of x as the result of a never-ending process; $0$ is finite and $n + 1$ is finite when ...
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Can a number have an uncountably infinite amount of digits?

I'm an extreme mathematical layman, so please excuse the probable ignorance and awkward phrasing of this question. Is there such thing as a kind of number which has an uncountably infinite amount of ...
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1answer
497 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
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Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
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Infinite sum of floor functions

I need to compute this (convergent) sum $$\sum_{j=0}^\infty\left(j-2^k\left\lfloor\frac{j}{2^k}\right\rfloor\right)(1-\alpha)^j\alpha$$ But I have no idea how to get rid of the floor thing. I thought ...
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One divided by infinity is not zero? [duplicate]

I know that $\frac{1}{\infty}$ is undefined. But my question is - can we say that $\frac{1}{\infty}\neq0$ ? I've got some idea how to explain that: Let's say we have a random-number generator that ...
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Does 1/∞ = 0.0̅01? Can some tell why not? [closed]

im just an 8th grade math student going into geomotry 1, but im curious to know if 1/∞ = 0.0̅01 (or 0.000...1). If that is so, then because 0.9̅9=1, then 0.0̅01 would be 0, thus proving that 1/0=∞ its ...
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Third axiom of Kolmogorov axioms

Let us define for a countably infinite set $S$ of real numbers that can be enumerated as $x_1,x_2,\cdots$, $$P(S) = \sum_{x \in S}p(x) = \sum_{i=1}^\infty p(x_i) = \lim_{n \to \infty}\sum_{i=1}^n ...
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How to find the limit of a sequence?

Question: If $0 < x < \frac{\pi}{2}$ and $f_k(x) = \tan(x)+\frac{1}{2}\tan(x/2)+ ...+\frac{1}{2^k}\tan(x/2^k)$. In Sigma Notation: $$f_k(x) = \sum_{n=0}^k \frac{1}{2^n}\tan\frac{x}{2^n}$$ ...
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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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2answers
93 views

$\ln{\left(\frac{1}{0}\right)} = -\infty$?

I have shown it using a theorem that I made, but I am not sure, as $\lim_{\alpha \to 0^{-}}{\left(\frac{1}{\alpha}\right)} = -\infty$, and $\lim_{\alpha \to 0^{+}}{\left(\frac{1}{\alpha}\right)} = ...
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Circle revolutions rolling around another circle

I just watched this video, and I'm a bit perplexed. Problem: ...
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W. Mückenheim claims a severe inconsistency of transfinite set theory; true? [closed]

The abstract for a paper on arxiv.org (http://arxiv.org/pdf/math/0408089v3.pdf) reads (with my emphasis): "Transfinite set theory including the axiom of choice supplies the following basic theorems: ...
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Are there more transcendental numbers or irrational numbers that are not transcendental?

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it ...
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Calculating $\lim_{(n,x) \rightarrow (\infty, 1^+)}x^n$?

Obviously, when $|x| \leq 1$, $\lim_{n \rightarrow \infty}x^n$ has a value such that $|x^n| \leq 1$, and $|x^n| \rightarrow \infty$ for $|x|>1$. However, this led me to wonder, what would happen ...
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Group action of $GL(2, F)$ on the projective line $P(F)$

I refer to section 8.3, page 119 of Algebra, A Computational Introduction by John Scherk. It is about group action of $GL(2, F)$ on the projective line $P(F) = F \cup \{\infty\}$. Given a matrix ...
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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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The probability of a number appearing in an approximation of an irrational number?

I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 ...
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Can we formally distinguish between actual and potential infinities?

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the ...