Questions tagged [infinity]

Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.

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Considering that there are an infinite number of numbers between two numbers, why infinite summation of equal terms cannot produce a finite result?

We know that for example the series 1/2+1/4+1/8+1/16+... does not get over 1. My reasoning of why this is possible, is that there are an infinite number of numbers or measurements between two numbers. ...
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How should one think of infinity.

I am currently taking a first year course in mathematics and have become a bit confused when confronted with "infinity". We are covering logic and sets and I had the following homework ...
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Dividing by square root of zero equals infinity?

So, my calculator app produced a result that doesn't seem correct to me. According to my calculator, $\frac{1}{\sqrt{0}}=\infty$. By my understanding, $\sqrt{0}=0$ (since $0^2=0$). So, shouldn't $\...
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-1 votes
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So in short terms, is infinity basically just describing a finite number? [closed]

What I'm asking is, wouldn't you agree that infinity is just describing a number we haven't named yet? Like numbers above the largest number that we know. We just stop counting because we have no ...
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Are there more real numbers in the interval $[1,\infty)$ than in the interval $(0,1]$? Or not?

We all might be familiar with the beautiful method Cantor devised to prove that the cardinality of the set of real numbers is more than that of the set of natural numbers (Refer to: https://en.m....
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Sum of infinitely cut lines

Suppose we split a line which length is $1$ in half. Then we get the $2$ lines which length is $1/2$ . Divide these two lines equally in half. Suppose we repeat this process infinitely. The length of ...
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Can we use mathematics and logic to estimate probability of extremely absurd events?

I'd like to detail my question over the example below. Let's say I have a random pixel generator which has $1024 \times 768$ screen resolution. It also has $24$ bit color which means $2^{24}= 16,777,...
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3 votes
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Infinite set such that sum of elements of every finite subset is not a power of $p$

Let $p$, be a prime number, and $S$ an infinite set of positive integers, such that all numbers from $S$ are coprime with $p$. Prove that there is an infinite subset $A\subseteq S$, such that for ...
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Can the continuity be established at $x= 0$ for the function below?

Is this method okay to show that $f(x)=x^{-2/3}$ is discontinuous at point $x= 0$? Limit $f(x)$ as we tends from positive side of $0$, we get $+\infty$ as the value, same with $f(x)$ approached from ...
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Why we ignored the $e^\infty$ term in the result? [closed]

In Question (2.20) of Griffiths' Quantum Mechanics book, they have given this Solution. In the Solution of question 2.20(b), they omitted $e^{(ik-a) \infty}$ (or may have considered $e^{(ik-a) \infty}=...
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Applying the "table proof" for uncountable numbers to countable numbers. [duplicate]

I was recently thinking about the Infinite Hotel Paradox and how it covers countable and uncountable infinities, and I was wondering what's stopping anyone from using the table proof for countable ...
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Cardinality between union of open uncountable set with finite set

I am trying to show that $|(0,1)| = $$|(0,1) \cup ${$2,3,4,5$}$|$ using the Schroder-Bernstein Theorem. Therefore, I need to find injections $f:(0,1) \rightarrow (0,1) \cup ${$2,3,4,5$} and $g:(0,1) ...
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1 answer
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The set of irrationals numbers is countable?

I tried to prove this using statement using the difference of sets $\mathbb{R}-\mathbb{Q}$ and the fact that $\mathbb{R}$ is not countable and $\mathbb{Q}$ is countable. In general, is it possible to ...
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Adding two infinite shapes to get itself?

This is at first a ridiculous question but on the other hand I'm not sure how to prove it. Take a solid 2D shape, $A$, and duplicate it to give $A'$. The shape $A$ together with the $A'$ make a new ...
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About set and interval [closed]

Every interval is an infinite set but every infinite set need not be an interval. A bit confused about the term
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Find $\lim_{n\rightarrow \infty}\frac{(\log n)^a} { n^b}$ when $ a,b >0$.

I need to prove that $(\log n)^a$ will always be smaller than $n^b$ as $n$ get larger (to infinity), also the condition is $a, b > 0$ ($a,b$ are random). I test the graph and it's true but I cant' ...
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Hilbert basis is this definition correct? Can't we get any simpler

There is a definition of Hilbert basis in this paper (chapter 1.2) that says : $(e_{i})$ is a Hilbert basis iff $$ (1). (e_{i}\dagger e_{j})=\delta_{i,j}$$ $$(2). \Sigma e_{i}e_{i}\dagger = I$$ where $...
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How can flipping a heads be 'almost surely' rather than an absolute guarantee?

If you flip a coin an infinite amount of times, where heads and tails are equal probability, then there exists some sequence of flips where all tails will show up (e.g. T,T,T,T...) . It's ...
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6 votes
0 answers
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Examples where mathematical objects "come down from infinity" or blow up instantaneously?

I recently learned about a surprising fact about the coalescent, which is a model used in population genetics to describe the genealogical relationships among individuals drawn from large populations. ...
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1 vote
2 answers
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How to understand the counterintuitive fact that you can calculate an infinite geometric series? [duplicate]

If you do additions and multiplications infinitely, it keeps increasing, doesn't it? And yet we have this formula (if 0<r<1) How to understand the counterintuitive fact that you can calculate ...
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relation between differentability and infinity norm of gradient of a function

Given a function $f$, do the following statements are equivalent? $f\ $ is differentiable. $\|\ \nabla f \|\ _{\infty} < \infty$. If so, can someone please let me know how it can be proved?
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Limits of General Infinite Series

tl;dr: How can I evaluate a limit if I don't have a good way to simplify the series? I was trying to evaluate the limit: $$L_1 = \lim_{n\rightarrow \infty}{{1 \over \sqrt 1}+{1 \over \sqrt 2}+{1 \...
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Hilbert's Hotel - kick everyone out

In Hilbert's Hotel scenario where an infinite number of new guests arrive, why can't we just kick everyone out and accommodate them again? Since there is already an infinite number of new guests ...
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How to prove of $\lim \frac{1}{|x|+1}=0$ as $x$ goes to $-\infty$?

I understand that $\displaystyle \lim_{x\rightarrow \infty} \frac{1}{|x|+1}=0$. Let $\varepsilon>0 $ and I choose $M>0$ such that $\frac{1}{M+1}<\varepsilon$. Thus, if $x\geq M$ then $$\left|\...
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Does: "an event recurs infinitely often almost surely" imply "the event occurs almost surely"?

I cite this textbook, chapter $6$. Some definitions: A measure preserving system is a probability space $(X,\Sigma,\mu)$ equipped with a measurable "dynamic" $\varphi:X\to X$ such that $\mu\...
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1 vote
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Is $\pi$ in the infinite set $ \{ 3, 3.1, 3.14, 3.141, 3.1415, ...\} $?

Does $\pi$ exist in the following infinite set. Apologies for lack of set notation, and i'm hoping its not necessary to help me understand the nature of infinite decimal expansion. I know sets are ...
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Simplifying the expression $\frac{ke^{ix}-ke^{ix}}{e^{ix}-e^{ix}}$

Say I have ended up with the expression: $\dfrac{ke^{ix}-ke^{ix}}{e^{ix}-e^{ix}}$. Let's rewrite it where $A = k$ and $B = e^{ix}$: $\dfrac{AB-AB}{B-B} = \dfrac{(1-1)AB}{(1-1)B}$. Again let's rewrite ...
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Union of uncountable set with cardinality c with a countable set

Let $A$ be an uncountable set with cardinality $\mathfrak{c}$ (so it is in bijection with the power set of $\mathbb{N}$) and let $B$ be a countable set (finite or infinite). Intuitively I want to say ...
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Are there any statements that are n-undecidable for all n?

I understand that in a sufficiently complicated, consistent formal system, not all statements are true, not all statements are decidable, not all statements' decidability is decidable, 3-decidable, ......
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Symmetry, homeomorphic sets, and the Cardinality of Even and Whole number sets being equal

Very similar questions to this have been answered on the following threads, but they have not answered this particular question about the symmetry of the relation used for calculating the cardinality. ...
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Functions which intersect asymptote $y=ax+b$ infinitely many times

I'm looking for a case similar to $f(x)=\frac{sin(x)}{x},\ y=0$. But I need a function $f(x)$ with the asymptote $ax+b,\ a\ne 0$.
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2 votes
3 answers
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Did the result of the Hilbert hotel paradox change after the proof $\mathfrak p=\mathfrak t$?

We have seen questions like What is the result of $\infty-\infty$ ? in 2011 and the result was that it is indeterminate. I find the example of $\infty-\infty=7$ absolutely convincing. We now (2016; I'...
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2 votes
2 answers
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When does $f(x) \sim g(x) + h(x)$ imply $g(x) \sim f(x) - h(x)$?

When does $f(x) \sim g(x) + h(x)$ imply $g(x) \sim f(x) - h(x)$? I tried to calculate: $$\lim _{x\to \infty }\frac{f-h}{g}$$ But was stuck at: $$ \lim _{x\to \infty }\frac{\frac{g}{g+h}}{\frac{f}{g+h}-...
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Closed form of equation with summation to infinity: brownian motion in a confined 1D box

I would like to implement this equation, which represent the brownian motion in one-dimension (1D) of a particle in a confined 1D box, which initial position is $x_0$ at time $t_0$ and final position ...
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2 votes
4 answers
421 views

What's the difference between a dense set and an uncountable set?

I once called a dense set an uncountable set. I was told this was wrong, as the set was dense, and not uncountable. I didn't have the mathematical knowledge to find this confusing, and instead thought ...
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1 vote
2 answers
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The Engagement paradox

Firstly, I should say that I came up with this paradox after reading of the Grimm Reapers paradox, but I’m not quite sure how this should be resolved. Nevertheless here is the problem: Suppose a lady ...
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How to find derivative of $\sum_{n=1}^{\infty}n$

I looked at another post for finding the derivative of another sum but it had $n$ on the upper bound. But what if you had $\infty$ on the top (as in my case)? I know I could find the derivative by ...
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1 vote
0 answers
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Can the entire range of 2 dimensions be represented in one dimension? [duplicate]

If you consider the x-y plane, any (x,y) point exists with x in -inf->inf and y -inf->inf. Is it possible to represent any x,y point in only one dimension (e.g. one number)? For example consider ...
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Point at Infinity

I started to take an introduction to complex analysis course, we learned open and closed regions in the complex plane, but our professor told us that the whole complex plane is open and closed at the ...
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1 vote
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prove that $ \int_0^{\infty}\frac{\{x\}\cos x}{x} dx \neq \infty $ [closed]

I can't prove that $ \int_0^{\infty}\frac{\{x\}\cos x}{x} dx \neq \infty $ ({x} — fractional part x). Calculations show that this integral is approximately 0.211... It suffices for me to prove that ...
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Is zero multiplied by does not exist equal to zero?

I watched a youtube video explaining how to get $\lim \limits_{x \to \infty} \left( \frac{\sin{(2x)}}{x} \right)$ He applied the properties of limits such that $\lim \limits_{x \to \infty} \left( \...
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2 votes
1 answer
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Determining the limit of a sequence [duplicate]

in my textbook there is a question that requires us to evaluate: $\lim_{n \to \infty } \left( \frac{10^{n}}{n!} \right)$ The solution solves this by evaluating: $\lim_{n \to \infty } \left( \frac{a_{n+...
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calculating mean of a CDF

I have this complex CDF: F(x)=1-exp(-X^2/c) when c is a constant. How can I calculate mean? In my calculation, I come to calculate this: ...
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5 votes
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Proving that limits at infinity are unique help

I am trying to prove that the $\displaystyle \lim_{x\rightarrow \infty}f(x)=L$ is unique, if it exists. Definition: Let $f:S\rightarrow\mathbb R$ be a function, where $\infty$ is a cluster point of $...
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0 votes
1 answer
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Suppose $A=\sum_{k=0}^{\infty}f(k)=B\prod_{n=0}^{\infty}g(n)$. If $A$ and $f(k)$ are known, how to find $B$ and $g(n)$?

My question like "some-to-product" or vice versa. See the following example (for reference, see here and here); $$\pi=\sum_{k=0}^{\infty}\frac{4(-1)^k}{2k+1}=2\prod_{n=0}^{\infty}\frac{4n^2+...
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0 votes
1 answer
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Is the Digit-Matrix in Cantors' Diagonal Argument square-shaped?

It seems to me that the Digit-Matrix (the list of decimal expansions) in Cantor's Diagonal Argument is required to have at least as many columns (decimal places) as rows (listed real numbers), for the ...
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1 vote
2 answers
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In general can we say $\infty=\infty$? Eg, $\sum_{i\in \mathbb{N}} i =\sum_{i \in \mathbb{Q}_+}i$

This might be a bit of a basic question but my current understanding is that we cannot. Still it makes me wonder if we can propose a mapping between two countable sets why not? For example why is this ...
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Question about a puzzle involving Euclidian numbers.

!!!Maybe still helpful but misguiding question|See answer of Doug M for insights|I mistook remainder for result. I hope you are doing well. So I did this course on Brilliant about infinities. It is ...
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A problem about decomposition of infinite set?

I am using lee larson's lecture note on introductory real analysis for self learning. And I am stucking in one problem which is like this: If $S$ is an infinite set, then there is a countably ...
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-1 votes
3 answers
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Is $2^{\aleph_0}$ well-defined?

Ok so I was thinking about power sets and mathematics that utilize infinity, and I ended up thinking about power sets for $\aleph_0$. Knowing that to get a power set you take $2$ and raise it to the $...
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