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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4
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5answers
2k views

why does commutativity of addition fail for infinite sums?

While discussing the sum of a particular series, $\sum\limits_{n=0}^{\infty}{\left(-1\right)}^n$ (a sum that I've heard is alleged to be equal to $\frac{1}{2}$), it was mentioned to me that addition ...
0
votes
2answers
57 views

Countably or Uncountably Many Discontinuities

I want to know why the following function has uncountably many discontinuities: $$f(x)=\left\{\begin{array} & x^2 & x \not \in \mathbb{Q} \\ 0 & \text{otherwise} \end{array}\right .$$ ...
0
votes
1answer
655 views

What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series ($A_n = 1 + \frac{1}{2} + ... + \frac{1}{n}$) and this ...
0
votes
1answer
56 views

map sum of square integers to a contiguous range of integers

Given a list $a$ of integers, $$n_{a_1}, n_{a_2}, ..., n_{a_d}$$ have $$N_a = \sum_{j=1}^d n_{a_j}^2.$$ The various $N_a$, $N_b$ etc. are integers, but are not contiguous: for example, if $d=2$, there'...
21
votes
7answers
6k views

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 ...
4
votes
1answer
1k views

Is there any mathematical or physical situations that $1+2+3+\ldots\infty=-\frac{1}{12}$ shows itself? [duplicate]

I just saw the proof that $$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts. I completely understood the proof and my question is NOT actually about the proof itself. At the end of the proof, ...
2
votes
0answers
543 views

An intuitive reasoning for 1+2+3+4+5… + ∞ = -1/12? [duplicate]

I was just watching this video: http://www.youtube.com/watch?v=w-I6XTVZXww In it, a professor working at the Nottingham University( Dr. Ed Copeland I think) shows how 1+2+3+4+5....+ ∞ = -1/12 Is this ...
4
votes
4answers
6k views

How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate]

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have ...
2
votes
3answers
441 views

Number of Infinities in complex numbers

How many infinities would be there for complex numbers? Like there are 2 infinities ($+\infty$ and $-\infty$) for the real numbers, is there a way to prove the number of infinities in the complex ...
4
votes
2answers
682 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
3
votes
1answer
80 views

Set theory, show a set is countable, homework. check my answer

I solved this question but there is something strange going on and I am unsure of myself. Would like someone to review it. We are given a total order (or linear order) $<^{*}$on group $A$ such ...
18
votes
3answers
861 views

What good is infinity?

I am becoming increasingly convinced that Wildberger's views are, if a little bizarre, at least not hopelessly inconsistent. When I was reading the comments in the video following (MF17), somebody ...
2
votes
1answer
82 views

Credit Given - Geometricly Modeling Infinity with 3 planes and 9 circles - Ratio of Circles

Refer to the attached diagram sketch to help visualize the equation. I am requesting help with an interesting math problem. Basically, I am diagraming infinity using three planes. These planes ...
4
votes
1answer
427 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
3
votes
4answers
208 views

random thought: are some infinite sets larger than other [duplicate]

I was in the shower today and I just thought of this so I'm asking it. I'm sure this has been thought of before. Let's say we have two sets, the set of all even numbers and the set of all natural ...
1
vote
3answers
104 views

How to evaluate this limit with l'hopital's rule

is it possible to use L'hopital for this or is there another method I'm missing? I have no idea how to even start this. $$\lim_{x\to \infty} \frac{(9x+1)^\frac12}{x+1} $$
1
vote
0answers
55 views

Number of real numbers between 0 and 1 vs number of all integers [duplicate]

Let a be the number of real numbers between 0 and 1 Let b be the number of all integers. Can I say ...
0
votes
1answer
39 views

information content of a quadratic surd

how much information is required to construct the equation: $$ X^2 - 2=0 \; ? $$ suppose, in a spirit of seasonal festivity, we squander a few further bits, and pamper ourselves with the additional ...
0
votes
2answers
34 views

Infinite expansion of non-linear expressions with 3 or more variables

I just realised that if we expand any of the non-linear expression with power of 3 or more we can't stop expanding them until we are dead. So for example: Expansion: $(a+b+c)^2 = a^2 + b^2 + c^2 + ...
3
votes
0answers
61 views

Cantor, longish lines and the Landau -o notations

in general terms this question is about the behaviour of functions of a real variable as their argument $\rightarrow \infty$. i will present the matter as concisely as i can, but my presentation will ...
2
votes
0answers
191 views

Fine-grained way to measure infinity

It is known that the cardinality of $R$ is equal to the cardinality of $R^2$, $R^3$, etc. But, intuitively these sets have different sizes. A possible way to formalize this intuition is to talk about ...
2
votes
2answers
95 views

Is probability meaningful in cases of infinity?

Is it meaningful to speak of probability in cases of infinity? For instance, consider me having an infinite line of balls arranged in the manner: - Red, Green, Blue, Red, Green, Blue, Red....... ...
6
votes
2answers
248 views

structure of the full symmetric group on a countably infinite set

trying to get a handle on the full symmetric group $S$ of permutations on a countable set $X$. i had never really thought much about this group, but now i look at it for the first time it appears a ...
1
vote
3answers
83 views

Limit Problem - No clue where to start

$$\lim_{x\to \infty}\frac{x^{2011}+2010^x}{-x^{2010}+2011^x}$$ I'm not sure where to even start with this one. One idea I had was that perhaps it could be "split" up as: $$\lim_{x\to \infty}\frac{x^{...
3
votes
1answer
457 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
0
votes
2answers
99 views

$f:\mathbb{R} \to \mathbb{R}$ be differentiable and $\lim\limits_{x\to\infty}f'(x)=1$, is $f(x)$ unbounded? [duplicate]

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a differentiable function such that $\lim\limits_{x\to\infty}f'(x)=1$,then is it true necessarily true that $f(x)$ unbounded? I think that it will always ...
3
votes
4answers
99 views

Is it valid to write $\displaystyle \lim_{x \to 0} \frac{1}{x^2} = \infty$?

AFAIK the limes of a term does not exist if that term does not converge, but I haven't found a suiting question here yet. This probably is a double of a similar question.
3
votes
5answers
543 views

Product of all primes

Is the product of all primes a natural number? In other words, is this true: $$ \prod\limits_{\text{primes}} p_i \in \mathbb{N} $$ And if so, what about just some of them: $$ \prod\limits_{\overset{...
4
votes
1answer
505 views

Are there different types of infinity? [duplicate]

Today in class my professor mentioned that there are different types of infinity. This confused me at first because I always thought infinity is just infinity. What are the different types of infinity?...
3
votes
4answers
137 views

Limit for $\lim _{n \to \infty}(n+2)^{2}\sin\frac{1}{n}$

Can't prove the limit $$\lim_{n \to \infty}(n+2)^{2}\sin\frac{1}{n}=\infty.$$ by definition it should start: Let $M>0$. There exists an $N>0$ for every $n>N$: $$(n+2)^{2}\sin\frac{1}{n}>M.$...
6
votes
2answers
351 views

On the Continuum Hypothesis

Let me start out by saying that I am not a mathematician. I read an article over at Scientific American that discussed the Continuum Hypothesis. I developed the following thought experiment that would ...
1
vote
5answers
122 views

Stuck with infinities

I have heard this "some infinities are bigger than others" . How can this be ? The context was that the cardinality of the set of integers is less than that of the cardinality of th real numbers , ...
1
vote
1answer
64 views

Question about $f$ continuous function with these conditions?

Suppose I have a differentiable and bounded function $$f: [0, + \infty) \longrightarrow \mathbb{R}$$ such that $$\forall x \in [0, + \infty) \, : f(x) \cdot f'(x) > \sin x.$$ The question is: ...
1
vote
1answer
103 views

Reducing a double summation with infinite limits

I've been solving a Renewal theory problem and I end up with this function $m(t)=e^{-4t}\sum_{k=1}^{\infty}\sum_{i=2k}^{\infty}\frac{(4t)^i}{i!}$. How do I solve or reduce the double summation? Is it ...
2
votes
1answer
1k views

What is infinity in complex plane and what are operation with infinity extended to complex numbers?

For a real number $a$, $$\infty + a = \infty,$$ and if $a$ is positive, $$\infty \cdot a = \infty$$ What is $\infty + a$ and $\infty*a$ if $a$ is non-zero complex number, where $\infty$ is real ...
2
votes
2answers
209 views

How can we define infinitary proofs?

In the first order logic the usual notion of a formal proof for a sentence $\sigma$ from a theory $T$ is a "finite" sequence ($<\omega$ - sequeance) of sentences which each one of them is a valid ...
1
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1answer
127 views

n-Ball Volume and surface with $n \rightarrow \infty$

I am thinking about something I just read: The volume of the n-ball is given by $V_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac n 2 + 1)}r^n$ and its surface area is $S_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac ...
1
vote
1answer
136 views

Aren't two infinite asymmetric graphs always identical?

Suppose you have an infinite graph $G$. I assume $G$ to be cubic and planar. No further conditions, so it will be asymmetric, maybe in the sense of cubic planar version of Rado's graph: Every possible ...
6
votes
3answers
353 views

$\frac{1}{\infty}$ - is this equal $0$? [duplicate]

I've seen that wolfram alpha says: $$\frac{1}{\infty} = 0$$ Well, I'm sure that: $$\lim_{x\to \infty}\frac{1}{x} = 0$$ But does $\frac{1}{\infty}$ only makes sense when we calculate it's limit? ...
2
votes
2answers
799 views

The proof of the infinity base of $\mathbb{R}^{\infty}$

We know that a finite basis of the finite-dimensional space $\mathbb{R}^n$ is $$ \{(1, 0, 0, 0,\ldots,0),\:(0, 1, 0, 0, 0,\ldots,0),\:(0, 0, 1, 0, 0, 0,\ldots, 0),\:\ldots,\:(0, 0, \ldots, 0, 0, 0, ...
4
votes
4answers
386 views

Hilbert's Hotel and Infinities for Pre-university Students

Hilbert's paradox of the grand hotel is a fun and exciting ground to base a talk on the set theoretic concept of infinity for interested students - even in middle- and high school. However, it does ...
0
votes
1answer
398 views

Proving the product of two series diverges to infinity.

Proving the product of two series diverges to infinity, given that one series (An) converges to a limit L and (Bn) diverges to infinity, I have to prove that the product of the two series (AnBn) ...
0
votes
3answers
237 views

Why does $0,\bar{9}$ equal $1$? [duplicate]

I am finding hard to understand why $0,99999..... = 1$ I have the following proof: Let $x$ be $0,9999...$ then $10x = 9,999...$ So $10x - x = 9,999 - 0,9999$ $9x = 9 \rightarrow x = 1$ From a ...
2
votes
2answers
637 views

Is an infinitely small percentage of infinity infinite?

I'm not a mathematician, but this question intrigues me: Is an infinitely small percentage or part of infinity infinite? Do the two infinities "cancel out", leaving you with a real number? It seems ...
0
votes
1answer
69 views

About the order of infinity of $\Re^n$

I would like to ask a question about the order of infinity of the $n$-dimensional space $\mathbb{R}^n$. I am not sure whether I use the appropriate notation/mathematical language or not - please ...
4
votes
1answer
227 views

Can we determine $A= 1!+2!+3!+…$'s digits starting from last?

After reading a bit about p-adic numbers, I came up with an idea. We know that for every natural number $k$, there exists a natural number $n$ so that for every $m>n$, there are at least $k$ zero ...
1
vote
1answer
70 views

Show A is countable infinity

One more question about set theory: $A\subseteq R$ is an infinite set of positive numbers. Assume there is a value $k \in Z$ such that for any $B \subseteq A$: $\sum_{i=0}^\infty b(i) \le k$ where $...
3
votes
3answers
406 views

Infinite compact subset of $\mathbb{Q}$

Can I find an infinite set in $(\mathbb{Q},\mathcal{T}_e|_\mathbb{Q})$ which is compact?
-5
votes
3answers
84 views

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$ [closed]

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$. How do you prove it?
2
votes
2answers
189 views

Limit Involving Factorials

How would you go about calculating $$ \lim_{x \to \infty} \frac{x!}{(x - k)!} $$ for some constant $k > 0$?