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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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1answer
25 views

What happens if we equate 0 to the right side in 1/0= infinity [on hold]

What happens if we equate 0 to the right side in 1/0= infinity ?
1
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4answers
68 views

Limit $\lim_{x\to\infty}\left(1-\frac{a^2}{x^2}\right)^{x^2}$

I found this example in a textbook, and I understand the author's reasoning and I also reached the same answer using L’Hôpital’s rule. However, I have two issues: Firstly: For any finite $a$, then ...
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0answers
36 views

Is the set countably infinite or uncounatble? [duplicate]

Prove that for all sets A if A ⊆ Z+ then either A is finite or |A| = |Z+|. Z+ means set of all positive integers
1
vote
1answer
27 views

Can I prove that a 2-variable limit does not exists if the limit on a curve is infinity?

Consider a 2 variable function $f(x,y)$ and the limit $$\lim_{(x,y)\to (0,0)} f(x,y)$$ If I find two continuous functions $\gamma_1(t)$ and $\gamma_2(t)$ such that $\gamma_1(0)=\gamma_2(0)=(0,0)$ ...
3
votes
2answers
69 views

Could one argue that $10 \cdot 10 \cdot 10 \cdot 10 \cdots$ is equal to 0?

The thing about this is, if we assign a variable: $$x=10 \cdot 10 \cdot 10 \cdot 10 \cdots$$ and then enclose all but one multiplicand in brackets (as multiplication is associative) and we get: $$x=10 ...
-2
votes
2answers
35 views

If $\lim_{n\to\infty}a_n=0$, then $\lim_{n\to\infty}a_n^n=0$ [closed]

Let $\{a_n\}$ be a sequence such that $\lim_{n\to\infty} a_n = 0$. Show that $\lim_{n\to\infty} a_n^n = 0$. I haven't been able to find anything about this question and would greatly appreciate if ...
0
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0answers
64 views

Is is possible to derive the axiom of infinity? [closed]

Is it possible to prove/derive the axiom of infinity (from the ZFC)? Perhaps through the use on logic?
1
vote
1answer
47 views

Test convergence of $n^{1/2} \sin\left(\frac{1}{n^{1/2}}\right)$ [closed]

Please help me show that $$a_n = \sqrt{n}\sin\left(\frac{1}{\sqrt{n}}\right)$$ converges to 1. By my understanding, it should be drawn to either $\infty$ or zero depending on which part "grows ...
4
votes
1answer
37 views

Is the concept of dimension still well defined for non-finite dimensional spaces? [duplicate]

The question is quite simple: if $\mathbb{V}$ is a vector space and $B$ and $B'$ are basis for $\mathbb V$, then do $B$ and $B'$ have the same cardinality? I've tried to answer the question as ...
0
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0answers
37 views

Can we state the existence of infinite set without infinity axiom? [duplicate]

I have a question about infinity axiom in ZF and maybe, it has nonsense. So I apologize in advance if it is the case. In ZF, the infiny axiom can be state as $\exists X(X\neq\emptyset\wedge\forall x\...
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0answers
21 views

how to find the asymptotic growth of f

The question is: Let $$S_n=\left\{t\in\mathbb{N}\mid t\text{ does not contain }n\text{ consectuive }4'\text{s}\right\}$$ E.g. $2464\in S_2$, but $2544$ is not. Let $$f(n)=\sum_{s\in S_n}\frac{1}{s}$$...
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votes
4answers
83 views

Evaluating the limit $\lim_{x \to \infty}\left(\sqrt{x^4 -x^3+1}-\sqrt{x^4+15x^2-5}\,\right)$

I wanna know how to do this limit $\lim_{x \to \infty}\left(\sqrt{x^4 -x^3+1}-\sqrt{x^4+15x^2-5}\,\right)$
4
votes
2answers
71 views

Is there a set in which division of 0 by 0 is defined?

The reason I ask this is that I've discovered that, even though they don't satisfy all field axioms, there are sets called projectively extended real number line and Riemann sphere, which are ℝ∪{∞} ...
1
vote
2answers
77 views

How would the set $\mathbb{N}= \{1, 2, 3, \dots \}$ be a finite set according to Definition 1.3.1 from Bartle's *Introduction to Real Analysis*?

This is from Sherbert and Bartle's Introduction to Real Analysis. 1.3.1 Definition (a) The empty set is said to have $0$ elements. (b) If $n\in \mathbb{N}$, a set $S$ is said to have $n$ elements ...
1
vote
3answers
105 views

Sum of all natural numbers.

Okay, I do know that there are three ways of showing that it equals $-1\over 12$: 1) The Reimann zeta function calculated for $-1$(see picture) 2) The one involving Grandi's series, the series $1-2+...
2
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1answer
25 views

Definitions of normalisers for infinite groups

If G is a group and A is a subset of G, the normaliser of A in G can be defined as either (1) $N_G(A) = \{g \in G\ |\ gag^{-1} \in A, \forall a \in A\}$ (2) $N_G(A) = \{g \in G\ |\ gAg^{-1} = A \}$ ...
0
votes
2answers
76 views

$(I-A)^{-1}=\sum_{i=0}^\infty A^i$

Let $V$ be a finite dimentional normed vector space and let $A$ a linear transformation from $V$ to $V$ such that $\left \| A \right \|<1$. Show that the linear tranformation $I-A$ is invertible ...
0
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0answers
22 views

Eigenvalues of an infinite matrix

I have matrix elements $$ H_{mn} = \begin{cases} \frac{1}{(m-n)^2} & m \neq n \\ 0 & m=n \end{cases}$$ If the indices extend to infinity what are the eigenvalues of this matrix? I'm sure ...
0
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3answers
38 views

Convince me: limit of sum of a constant is infinity

So I have a problem and have simplified the part I am confused about below. If $\sum_{m=1}^{\infty }c < \infty$ and $0 \leq c \leq 1$, then $lim_{n\rightarrow \infty} \sum_{m=n}^{\infty }c= 0$ ...
0
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2answers
47 views

Additive inverse of infinity [closed]

What example demonstrates that a definition $$ \infty-\infty=\infty+\big( -\infty \big)=0 $$ necessarily invokes a contradiction?
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4answers
40 views

Isn’t dividing by 0 similar to multiplying by infinity?

A commonly cited proof for being unable to divide by zero is as such: 0 = 0 * 1 0 = 0 * 2 0 * 1 = 0 * 2 (divide both sides by 0) 1 = 2 That’s obviously ...
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0answers
25 views

Limit of infinity to the power of infinity vs factorial

I've been playing around with Taylor polynomials for sin(x) and it makes sense that the polynomial series converges to sin(x): For x as a reasonable number, the ...
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votes
2answers
50 views

What is the correct answer of 1/0 or 1÷0 and why? [closed]

Few days ago, i was attending lecture of Introduction to Computers (ITC) in my University and there was one question. ** What is 1/0 or 1(divide-by) 0. ** I checked it on my phone and it says 1/0 ...
3
votes
2answers
102 views

Why this random walk can't go on forever?

$A$ starts with $i$ coins, $B$ with $N-i$. At each trial, $A$ gives one coin to $B$ with probability $p$ or $B$ gives one coin to $A$ with probability $q$ where $p+q=1$. This can be modeled as a 2D ...
2
votes
3answers
51 views

Orthonormal basis: Countable $\infty$ vs. Uncountable $\infty$

My doubt is the following, when you create an orthonormal basis for a space, the number of coefficients in each vector, and the number of vectors is equal to the dimension of the space (at least in ...
1
vote
3answers
31 views

Limit at infinity $\lim_{x\to \infty} x^a a^x=$?

$\displaystyle \lim_{x\to \infty} x^a a^x=?$; $0<a<1$ I try to use the property: $a^{\log_a x}= x$ and reescribe the expression $\displaystyle \lim_{x\to \infty} x^a a^x = \lim_{x\to \infty}...
1
vote
1answer
48 views

Do These Primitive Definitions of Convergence Over the Natural Numbers Appear in Greek Mathematics?

We use the notation $\mathbb N^{\gt 0} = \{1,2,\dots,n,\dots\}$. If $m,n \in \mathbb N^{\gt 0}$ we can always apply Euclidean division to get a quotient - if $m \ge n$ we can call $m$ the dividend ...
1
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2answers
43 views

Would the sum of an infinite series be finite if there were gaps between each numbers that increased until reaching infinity?

If you were to have a series: 1+1+2+4+7+11+16+22... Where the gap between the numbers increased by one every time, wouldn't this gap eventually become infinite? Once it was, what would happen? Would ...
1
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2answers
71 views

Problems with Cantor's diagonal argument and uncountable infinity

Cantor's diagonal argument: As a starter I got 2 problems with it (which hopefully can be solved "for dummies") First: I don't get this: Why doesn't Cantor's diagonal argument also apply to natural ...
0
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2answers
67 views

Proof of d'Alembert's ratio test: for sequences tending to infinity

What would a proof for this theorem look like? Suppose that $(a_n)_n$ is a sequence such that $a_{n+1}/a_n$ tends to $\ell$. Prove that if $\ell>1$ and $a_n>0$ for all $n$ in the naturals, ...
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1answer
41 views

Archer duel problem

Two archers take alternating turns trying to hit the opposing archer. As soon as the shooting archer hits the other, the shooting archer wins. Both have a specific probability p to hit the other. p1 ...
1
vote
1answer
62 views

radially unbounded functions

Is the following function radially unbounded or not? $$V(x) = \frac{x_{1}^2}{1 + x_{1}^2} + x_{2}^2$$ I know that if $x_{2} \to \infty$ in which case $||x|| \to \infty$ and $V(x) \to \infty$ but if $...
0
votes
2answers
38 views

Number having sexagesimal expansion end with infinitely many zeros?

I am looking for all the real numbers whose sexagesimal expansion (base $60$) ends in infinite tail of zeros. Does they really exist? It seems absurd to me or mm thinking it in a wrong manner?
1
vote
1answer
28 views

Why do we need the following set to have an infinite set?

By Dedekind's definition: Definition: We say that some set $S$ is infinite if there exists an injection $f:S\rightarrow S$ such that $\operatorname{Im}(f)\neq S$ (or $f(S)\neq S)$. Now, the book ...
0
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0answers
23 views

Prove that this $ \sup $ is finite

How can i prove that this supremum is not $ \infty $? $$ \sup_{\vert\xi\vert\geq 1} \left( \dfrac{\left(1 + \sqrt{4\vert\xi\vert ^{2}-1}\right)^{i}}{\vert\xi\vert ^{i-1}\sqrt{4\vert\xi\vert^{2}-1}} \...
1
vote
1answer
34 views

Proving that $(a,b)$ is $F_{\sigma},\forall a,b\in\mathbf{R}$

I am required to prove that the interval $(a,b)$ is a $F_{\sigma}$-set i.e. it can be written as a union of countably many closed sets in $\mathbf{R}$. The following is my attempt so far. I did ...
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votes
1answer
32 views

Why following the below conditions, the $\sum_{i=0}^{\infty} a_n*b_n$ in converge? [closed]

Given $a_n$ a positive series and $\sum_{i=0}^{\infty} a_n$ is converge, and given also that $\lim_{n\to \infty} b_n = 100$, then the summation $\sum_{i=0}^{\infty} a_n*b_n$ is converge.
0
votes
1answer
47 views

Is it possible to reduce an algebraic function to $1$s and $0$s?

I want to know where in this process I'm going wrong. Perhaps it's not even a valid thing to do...? Take a well-behaved function such as $f(x)=x \sin 2x$. I want to turn this into a new function $g$ ...
1
vote
2answers
77 views

Why the $\lim_{n\to\infty} (\frac{n}{n^2+1}+\frac{n}{n^2+4}+…+\frac{n}{n^2+n^2})= \frac{\pi}{4}$?

Why the $\lim_{n\to\infty} (\frac{n}{n^2+1}+\frac{n}{n^2+4}+.....+\frac{n}{n^2+n^2})= \frac{\pi}{4}$? I read somewhere that it is related to $f(x)=\frac{1}{1+x^2}$ but dont know why...
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votes
1answer
34 views

why this statement about $\sum_{i=0}^n a_n$ is false?

Given that ${a_n}$ is positive series, and $\sum_{i=0}^n a_n$ is converge: -There is a sub-series for ${a_n}$ that converge to S>$0$. (THIS statement is false) why is this statement false?
0
votes
1answer
35 views

$\sum_{n>k}\frac{j}{j+1}\cdot (j+1)^{-n}=\frac{j}{j+1}\cdot (j+1)^{-(k+1)}\cdot \sum _{n=0}^{\infty}(j+1)^{-n}$

Given $1\leq j\leq 6$ and $k\in \mathbb{N}$, why is this equality holds? $\sum_{n>k}\frac{j}{j+1}\cdot (j+1)^{-n}=\frac{j}{j+1}\cdot (j+1)^{-(k+1)}\cdot \sum _{n=0}^{\infty}(j+1)^{-n}$
3
votes
1answer
59 views

Over ZF, does CUCSCS imply that every infinite set is Dedekind-infinite? (C: Countable, etc.)

We add Axiom CUCSCS: The countable union of countable sets is a countable set. to ZF. Is every infinite set now Dedekind-infinite? My work: When I look at CUCSCS I see no natural path of building ...
0
votes
2answers
79 views

Why is $\cos(\infty)$ undefined?

Why is $\cos(\infty)$ undefined? I really don't understand this. Is it because we can't pinpoint an exact value for cos at infinity?
3
votes
3answers
54 views

Why is the homogeneous line through all points at infinity (1:0:0) and not (0:0:0)?

So I just had a geometry lecture that introduced me to homogenous coordinates. To be clear with notation let me recap: Homogenous coordinates in $\mathbb R^n$ space are described as $$(x_0:x_1: ... : ...
4
votes
3answers
64 views

Show that $\lim_{x \rightarrow \infty} f'(x)<1$ implies $f(x_0)<x_0$ for some $x_0$

Let $f:[0,\infty)\rightarrow R $ be a continuously differentiable function. Show that if $ \lim_{x \rightarrow \infty} f'(x)<1 $ then $ f(x_0)<x_0 $ for some $x_0$ large enough. (An ...
0
votes
1answer
58 views

How does Cantor's diagonal argument actually prove that the set of real numbers is larger than that of natural numbers? [duplicate]

I have looked into Cantor's diagonal argument, but I am not entirely convinced. Instead of starting with 1 for the natural numbers and working our way up, we could instead try and pair random, ...
-1
votes
2answers
40 views

Random selection in an infinite set

If I had an infinite number of sticks and (somehow) painted sticks #1,4,7,10.. in red and then painted sticks #2,3,5,6,8,9... in blue. Then I picked a stick at random, do I have more chance of ...
0
votes
0answers
34 views

Differential equation with infinite initial value

I know that given an initial value problem \begin{eqnarray} f'(x) &=& G(x,f(x)) \\ f(0) &=& c \end{eqnarray} there is a unique solution for $f$ for any sufficiently nice $G$ (in this ...
1
vote
1answer
62 views

What does this ZF+[AOC-Lite] Look Like?

We use the notation $[n] = \{0,1,2,3,\cdots ,n-1 \}$. Remove AOC completly from ZFC and then replace it with Axiom asdf: Let $X$ be any nonempty set such that $\;\text{For every injective } f:[n]\to ...
2
votes
1answer
56 views

Devils and Infinity [duplicate]

You are in hell for all eternity and the devil gives you two dollar bills every day with increasing serial numbers. He then takes the dollar bill you have with the smallest serial number. At the end ...