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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How to calculate the $n$-th member of sequence $a_{n+1}=\sqrt{y+a_{n}}$

I was searching for function that following: $a_{x+1}=\sqrt{y+a_{x}}$ and $a_{0}=0$ and I found only for $y=2$ or $y=0$. For $y=2$: $f(x,2)=2cos(\frac{\pi}{2^{x+1}})$. For $y=0$: $f(x,0)=t^{2^{-x}}$ ...
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1answer
39 views

The real projective line and $1/\infty$

so I came up with this idea: the real projective line defines that $\infty = - \infty$. What if I divide any value $x$ (not equal to $\infty$) by infinity? Would that be 0? or "something" between ...
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8answers
2k views

After switching a lamp on and off infinitely many times in one minute, is it on or off? [duplicate]

So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another ...
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2answers
36 views

Possible proof for the multiple sizes of infinity [on hold]

One can easily relate all regular polygons to the triangle: triangle = 1 triangle square = 2 triangles pentagon = 3 triangles hexagon = 4 triangles and so on and so forth… A circle is basically ...
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5answers
124 views

What we get if we add 1/2 infinite times [on hold]

I want to know if this is correct We have this sums: $$S1=1-1+1-1+1-1+1-1+1-1...=\frac12$$ $$S2=1-2+3-4+5-6+7-8...=\frac14$$ $$S3=1+2+3+4+5+6+7+8...=-\frac{1}{12}$$ If we take ...
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7answers
144 views

The meaning of the symbol $\infty$ in Spivak's calculus book

Spivak in "Calculus" writes ... symbols of $\infty$ and $- \infty$ are purely suggestive: there is no number $``\infty"$ which satisfies $\infty \geq a$ for all numbers $a$. What is the meaning ...
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2answers
96 views

How does Infinity really work, and the relation with ∞ and space [closed]

Here is my question. In math, everyone always says $∞$ is a number, but you can't count to it. Is infinity just continuous generation of numbers, or is it actually a number that means numbers just ...
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2answers
64 views

I am trying to find the limit of P(x)

When I am looking for a $\lim\limits_{x \to -1} P(x)$ where P(x)$= \sum \limits_{n=1}^\infty \left( \arctan \frac{1}{\sqrt{n+1}} - \arctan \frac{1}{\sqrt{n+x}}\right) $ do I have to ignore a ...
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4answers
697 views

Alternate method to calculate an infinite string of numbers that's not $\pi$, and contains any string

So, rather than using $\pi$, is there any way that isn't overly complicated, (and can be calculated on a computer without taking a year) in which I could generate an infinite string of numbers that ...
3
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3answers
162 views

Infinity indeterminate form that L'Hopital's Rule: $\lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}}$

When I tried to find the limit of $$ \lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}} $$ by applying L'Hopital's Rule the order of denominator would increase. What else can I do for it?
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1answer
45 views

How to show uniform convergence of series

Let $$f(t) = \sum_{k=0}^\infty ke^{-t\sqrt{k}}u_k$$ for $t \in (0,\infty)$, where the $u_k$ is such that $\sum \sqrt{k}u_k$ converges, but we know nothing about the convergence of $\sum ku_k$. How do ...
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2answers
39 views

How much information does learning this interval give you?

Let's say you have a number $x$, and a priori, you know that $x \in [0, 1)$ (each value from 0 to 1 is equally likely.) Then a wizard comes and tells you that $x \in [a, b) \subseteq [0, 1)$. How much ...
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2answers
40 views

Complex infinity when proving divergence

My calculus course book (Adams' Calculus) does not explain why $(-1)^n$ diverges (it just says "$(-1)^n$ simply diverges"), and I tried to see why it diverges by taking its limit as $n$ approaches ...
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2answers
48 views

Mandelbrot Set area

If there are an infinite amount of details that can be found in a Mandelbrot set, shouldn't the Mandelbrot Set have an infinite area? Supposedly the area of a Mandelbrot set is 1.5065918849 ± ...
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5answers
2k views

Are there fewer reals on $(0, 1)$ than on $(1,\infty)$?

I know that the cardinality of the sets of real numbers $(0, 1)$ and $(1, \infty)$ are equal. So what is the fallacy in this argument? For every real on $(0, 1)$, we can add any integer $n$ to it ...
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25 views

Why ∞-∞ is unspecified value? and why ∞/∞ is unspecified value? [duplicate]

1- Why ∞-∞ is unspecified? The answer 2- Why ∞/∞ is unspecified? The answer
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1answer
25 views

If $\dim(V_F)$ is Infinite, Does It Follow $\dim(\operatorname{Hom}(V_F, W_F)) \ge |F|$?

Part of the proof that $\dim(V^*_F) > \dim(V_F)$ for an infinite dimensional space is that $\dim(\operatorname{Hom}(V_F, F)) \ge |F|$ (i.e $\dim(V^*_F) \ge |F|$). See for example Dual space ...
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6answers
99 views

More numbers between $[0,1]$ or $[1,\infty)$?

There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are ...
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1answer
55 views

Improper integral: why $\int_0^1(x^2+ x^{1/3})^{-1}\,dx$ is convergent and not $\int \frac{1}{x^2}\,dx$ ???

How do I show that $\int_0^1(x^2+ x^{1/3})^{-1}\,dx$ converges? I assume you show it on $(0,1]$. Can't seem to get my head around why this would be true.
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31 views

For Vector Spaces V and W with one Infinite Dimensional , is Hom(V, W) Isomorphic to Hom(W, V)?

If V and W are both finite then clearly Dim (Hom(V, W)) = Dim(V).Dim(W) = Dim(Hom(W, V)) so they are isomorphic. I'm not so sure if one is infinite. An "infinite matrix" construction for a linear ...
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2answers
60 views

What are some good reasonably rigorous texts on the mathematics of infinity?

The Infinite Book is too light and not focused enough on the mathematics of infinity, and Everything and More: A Brief History of Infinity has too much focus on the history of infinity instead of the ...
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2answers
40 views

What is the correct mathematical notation for something comprised of the sum of constituents n where n is infinite?

I am trying to figure out what the correct mathematical notation would be for something like the following: I want to describe that the value V of a company is equal to sum of parameters P at any ...
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2answers
53 views

Is my argument correct to solve this textbook problem?

The problem is from M.Bona's "A Walk through Combinatorics", Ch1 Prob 13: There are infinitely many pieces of paper in a basket, and there is a positive integer written on each of them. We know ...
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3answers
59 views

Find a limit without using L'Hopitals rule 9

Can someone please show me how to do this without using L'Hopitals rule: $$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x$$ I know the limit is $e^a$, but I would like to know the steps taken to ...
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1answer
38 views

What is the limit of the below functions when n tends to inifinity?

What is the value for the functions in the image when limit n tends to infinity?. Also what is the asymptotic complexity (big $O$ notation) for all the four functions?. $$\begin{aligned}f_1(n) &= ...
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2answers
106 views

derivative of x^x^x… to infinity?

I am a 12th grade student, and I am afraid that in realistic terms this question might not even make sense because of the infinities that have to be dealt with. However, in my attempt to calculate ...
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3answers
68 views

Why does $e^{\frac10}\neq e^{\frac1{-0}}$?

I was unable to explain why this fails? I asked to it many peers and they too can't. I faced this situation when solving a kind of integration problem. Consider $x=-x$ Then $x=0$ That is, $0=-0$ ...
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49 views

Which has greater powers: $0$ or $\infty$? [duplicate]

Okay, I know that this may seem silly, but please try to clear my doubts. The question is, that which 'term' has more powers. If we multiply $\infty$ and $0$, what do we get. I know that infinity is ...
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3answers
221 views

Do different infinities have different base representations?

I've been wondering about different types of infinity; e.g., $\aleph_0,\aleph_1$, e.t.c.; where $\aleph_0$ represents the smallest infinity, the countable infinity (e.g., the cardinality of the ...
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3answers
81 views

Proof of $\sum_{x = 1}^\infty \frac{1}{x}$'s divergence by absurdity?

(From this site.) The following argument purports to show that the series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \dots = 0$. It begins with the harmonic series. $$ \begin{aligned} \sum ...
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1answer
144 views

determinant of infinitely large matrix by decomposition

Read the too long didnt read version in bold before going into the finer detail. The overall point is that when I decompose this matrix to try and find its determinant I get an answer that doesn't ...
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0answers
72 views

0's reciprocal (Theoretical) [closed]

Background: I am in 8th grade and I like to study around advanced mathematical subjects. However, I do not know enough to be sure in my conjectures. Therefore, I would like your help. I have a ...
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2answers
42 views

Difference between intersection of infinite sets having finite, and having infinite elements

I could find individual answers for both of these, but can someone compare how being a finite set or an infinite changes the final outcome? (a) If A1 ⊇ A2 ⊇ A3 ⊇ A4 · · · are all sets containing an ...
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2answers
157 views

Rolling two dice, what is the probability that two consecutive $7$s happens earlier than a $12$?

Alice and Bob are playing a game involving two dice. If a sum of 12 appears, Alice wins and they stop playing. If a 7 appears twice in a row, Bob wins and they stop playing. What is the probability ...
2
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5answers
255 views

Is there an object in reality that is proven to be uncountable? [closed]

I've always wanted to come up with a fairly concrete example of an object or realistic set that could be uncountable. Most of the sets I can think about, even the hugest ones, are always countable. ...
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3answers
217 views

Why does Wolfram Alpha say that $n/0$ is complex infinity?

I typed a number divided by 0 on Wolfram Alpha and thought that it would say "undefined". However, when I pressed enter it told me that the answer is complex infinity. I have always been taught ...
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3answers
84 views

Trying to show that $\sum_{1}^{\infty} \frac{n^n}{n!} $ diverges.

I have been trying to prove this using the ratio test $|\frac{A_{n+1}}{A_n}|$ , which leads me to this expression: $$\left|\frac{(n+1)^{n+1}}{(n+1)!}\cdot ...
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1answer
60 views

Evaluate the limit $\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $

It seems reasonable to assume that $$\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $$ goes to zero but I can't figure out how to prove it.
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1answer
49 views

Infinite solutions for $(\frac{n+1}{n})^a\cdot (\frac{m+1}{m})^b = 2$

Given $(\frac{n+1}{n})^a\cdot (\frac{m+1}{m})^b = 2$ where a, b, n, and m are all positive integers, are there infinitely many solutions $(a,b,n,m)$?
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109 views

Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ [duplicate]

I'm completely stuck evaluating $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ how would I go about solving this?
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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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5answers
94 views

To evaluate $\lim_{n\to\infty} \dfrac{10^n}{n!} .$

I am having a lot of trouble evaluating $$\lim_{n\to\infty} \dfrac{10^n}{n!} .$$ I know intuitively that $n!$ is much larger and this expression should go to zero but I just can't figure out how to ...
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2answers
35 views

recursive sequences bounded above and their limits at infinity

Define a sequence $\langle a(n)\rangle$ recursively by $a(1)=\sqrt{2}$ and $a(n+1)=\sqrt{2+a(n)}$ $(n>0)$. a)by induction or otherwise show that the sequence is increasing and bounded above 3. ...
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1answer
35 views

bounded sequences and limits at infinity

Show that if ${a_n}$ and ${b_n}$ are sequences for which $\displaystyle\lim_{n\to\infty}a_n=0$ and ${b_n}$ is bounded, then $\displaystyle\lim_{n\to\infty}a_nb_n=0$ Sorry I'm on my phone and I'm not ...
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2answers
64 views

Probability over an infinite set. Of two numbers.

Suppose there is a man, who chooses two completely random numbers. So they can be equal too. And they can be only positive real numbers. One of them can even be $285294.38285967281$ or anything ...
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55 views

infinite series and proof of sum using induction.

Consider the series: $$ \sum_{i=1}^\infty \frac{i}{(i+1)!} $$ Make a guess for the value of the $n$-th partial sum and use induction to prove that your guess is correct. I understand the ...
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2answers
56 views

Infinity - infinity calculus

$$\lim_{x\to\infty} (x-1)e^{-1/x}-x$$ I know that this limit equals $-2$ but I don't know how to prove it. I can only get to $\infty-\infty=?$
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31 views

Are there any certainties of properties of infinity?

I have been considering the properties of infinity and applications to various areas of maths and was hoping to get some opinions of more seasoned mathematicians than myself. One geometric ...
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1answer
40 views

Limit of functions involving trigonometry as n approaches infinity

By graphing these functions, I know that P(n) approaches pi as n tends towards infinity. However, is there a mathematical way for proving this? I am doing a maths exploration on Archimedes' ...
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2answers
48 views

limit as $x$ approaches infinity of $\frac{1}{x}$

How can I show $\lim_{x \to \infty} \frac{1}{x} = 0$ using epsilon delta proof. Its pretty obvious that the limit is zero, but I am still new at epsilon proofs.