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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Can $f(\infty)$ be defined if the sequence $f(n)$ is divergent?

Let there be given an real-valued function $f(n)$ , with $n\in\mathbb{N}$ , $a,b \in\mathbb{R}$: $$ f(n+1) = a f(n) + b $$ What then is the value of $f(\infty)$ ? As a physicist by education, being ...
0
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1answer
18 views

Series having strict inequality implies limits having strict equality?

I was wondering if I have two convergent series, say, $\sum_{n=1}^{\infty} s_n = s$ and $\sum_{n=1}^{\infty} t_n$ = t, and for all their partial sums we have that: $s_n > t_n$. Is it then ...
4
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1answer
90 views

Apples and Infinity

I am taking a proof writing and discrete mathematics course and we are learning about infinity. My TA asked me the following question and I'm wondering if my solution is correct? Question:Suppose ...
36
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3answers
2k views

Why can you chose how to align infinitely long equations when adding them?

I saw in a video this proof: Take this equation: $$f=1+\frac12+\frac14+\cdots$$ and do this: $$\begin{align} f&=1+1/2+1/4+\cdots\\ -\quad f/2&=\quad\:\:\:1/2+1/4+\cdots\\ ...
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2answers
59 views

what is the integral on $[0,2]$ of $x/(3-2x)$

i know that this is an improper integral, but when you evaluate the limits as $x\to (3/2)^-$ and $x\to (3/2)^+$, you get positive and negative infinity but I am not sure if you can cancel them ...
0
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2answers
48 views

Help with an improper integral!

I need some help with an indefinite integral problem (only the $2^{\textrm{nd}}$ part thou). Problem is as follows. Consider the function $f(x) = \dfrac{\ln\!\left(x\right)}{x^{p}}$, where $p>1$ ...
2
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1answer
74 views

Geometric definitions of infinity

There are several definitions of "infinite set" that are common in set theory. For instance, a set $S$ is finite if there is a bijection between a natural number $n$ and $S$; it is infinite ...
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3answers
497 views

How to prove the infinite number of sides in a circle?

I was in geometry class today when I came across the following formula for the external angle of a regular polygon with n sides: $$Ea = \frac{360º}{n}$$ So I thought if $$ n\rightarrow\infty $$ then ...
0
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0answers
27 views

Mapping from 'one infinity to another'.

I've been wondering about this for a while now. Consider the function $f(x)=\frac{1}{x}$. When $x\in(0,1)$, the function maps to the interval $(1,\infty)$. Conversely, anything in the interval ...
0
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1answer
111 views

Infinity equals zero

I am not a math whiz by any degree, so go easy on me if I've made a huge oversight. I am working on a geometrical proof for a philosophy project I am working on and I need some help with its current ...
0
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1answer
31 views

limit of a floor function

I came across the following limit in a math book $\lim_{x\to \infty}\frac{(x^x)}{E(x)^{E(x)}} $ where $E(x)$ represent the floor function, and the question was to prove that this limit doesn't ...
2
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3answers
66 views

If an object halves its speed every second (but never gets to 0), will it eventually get from point A to point B?

There is a ball that starts at point A on a line and moves toward point B. Every second, it moves half of the distance left, but never stops moving: Etc. Would the ball ever reach point B? In one ...
0
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4answers
40 views

infinity sum of the fractional

Can anyone explain how to simplify $ \frac{2}{3} + \frac{6}{9} + \frac{12}{27} + \frac{20}{81} + \frac{30}{243} + . . . $ I have no any idea since i dont have pattern i can't do it with integral or ...
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0answers
29 views

How do I show that infinite application of this function gives a constant?

I want to show that g(x) returns the same value independent of x and hence is a constant. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
0
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1answer
47 views

Find the equation of a line parallel to the y-axis, that goes through the point $(\pi,0)$

I have been trying to do this problem and I am very confused. I know the gradient is infinity when any line is parallel to the y-axis, therefore, $y = \infty \cdot x + c$, right ($y = mx + c$ being ...
0
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2answers
54 views

Reversing a number of infinitely many digits

Lets say we have a function that gets as input a real number and returns its reverse e.g. 123.12 -> 21.321 So what happens when the input is a number α that has infinitely many digits. Does then ...
0
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1answer
96 views

Why doesn't this proof work? (Zero times infinity equals zero)

Say we are trying to prove $$ 0\cdot n = 0 $$ By mathematical induction, we start with a base case of n = 1 $$ 0\cdot 1 = 0 $$ So now we assume our original formula is true, and try to prove a case ...
3
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1answer
60 views

Is the integral of the sum really the sum of the integrals?

I was asked to find the mclaurin series of $\int_0^x\frac{\arctan (t)}{t}dt$ using the known mclaurin for arctan: $\arctan(t)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}t^{2n-1}}{2n-1}$ Ok, so what I did ...
0
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1answer
91 views

Does an unbounded straight line have infinitely many axes of symmetry?

In a circle, any diameter is an axis of symmetry, so technically a cirle should have infinitely many axes of symmetry. This got me thinking about the axes of symmetry of straight lines. A line ...
0
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0answers
22 views

Convention on infinity comparision

What's the difference between $-\infty \leq a \leq \infty$ and $-\infty < a < \infty$ conceptually or otherwise. It doesn't really affect the solution to the solution of this problem I'm ...
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1answer
26 views

Calculating limits using the definition of number e

I have some examples in Demidovič using this technique and there seems to be no reliable source for them online, so I'll make a small tutorial. Example 1: ...
0
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1answer
77 views

DTFT of Impulse train is equal to 0 through my equation.

Let me have an impulse train function as below, $$ x[n] = \sum_{m=-\infty}^{\infty} {\delta[n-f_0 m]} $$ where, $f_0 \in \textbf{Z}$. Now, I am trying to calculate its DTFT, so I put it into DTFT ...
2
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3answers
193 views

What is the meaning of “uncountably infinite” within countable models of set theory?

So, I don't know much about countable models of set theory, other than that they exist. To me, their existence is a very weird thing (and a reason to move away from first-order formulations). Here is ...
0
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3answers
79 views

Filpping a coin infinitely many times

If a coin was flipped an infinite number of times, is it guaranteed to be heads at least once?
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4answers
57 views

Real analysis - Countable and uncountable set

I'm having a problem understanding this: The union of a countable set and an uncountable set is uncountable. Help me please!
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1answer
118 views

The sum of $1+1+1+1+…$

My teacher recently showed me a rather weird result and I would like to know if he was just tricking me or if he was serious. He showed me that $g=1-1+1-1+1-...=\frac{1}{2}$ Then he said that ...
3
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0answers
74 views

Suppose some theory T has countably many axioms, how many models of $T$ are there of cardinality $\aleph_1$,$\aleph_2$,$\aleph_{\omega_1}$?

Setting Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes. So we see $T$ has ...
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1answer
53 views

Evaluating $\lim_{x\to -\infty} \frac{(x-1)}{(x^{2/3}-1)}$

The limit at negative infinity should not exist, right? $$\lim_{x\to -\infty} \frac{(x-1)}{(x^{2/3}-1)}$$ for positive infinity, the limit is infinity, but the function is undefined for values less ...
1
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0answers
53 views

Cantor set countable? [duplicate]

I know the Cantor set is uncountable, but I just came with an argument that shows it is countable. Obviously my argument is wrong, but I just don't know where is the mistake. Here it is. Let $C$ be ...
0
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3answers
105 views

Discount in Infinity

I fully understand that $0.9999...$ mathematically will equal 1.0 exactly as the repeating decimal continues infinitely. This would be contrary to conventional logic where such a fraction is ...
0
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1answer
74 views

Can Aleph Numbers be multiplied?

i.e., does it make sense to say something like $(2 * \aleph_0) > \aleph_0$ ? The original question I was thinking about is: if A = $\mathbb{Z}$ and B = {the set of even integers} is it correct to ...
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10answers
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Is there a maximum value between open (0,1) set?

This question came up in my interview for a job application(you won't believe it but it was a C# programmer job application). Let's say we have a open set (0,1). Can we say that there is a maximum ...
4
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1answer
87 views

How to make a good “infinity plot”?

This is what I mean (note the labels in the x axis): The reason I'm looking at this problem is because I've always felt something was not right with truncated plots (e.g. of the exponential ...
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2answers
45 views

Indeterminate form as a series

We know that $0 \times \infty$ is an indeterminate form. However, is it equivalent to $0 + 0 + 0 + \cdots$? If yes, why we do not consider $\displaystyle \sum_{n = 0}^\infty 0$ an indeterminate form? ...
4
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1answer
58 views

Is the spacing between the set of all natural number powers bounded?

I was wondering whether the set of all numbers that can be expressed as a natural number to the power of another natural number has "infinitely wide" gaps or if there is some upper bound between the ...
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4answers
161 views

What's wrong with this “proof” that $\infty = -1$?

So I'm not great at math which is why I'm asking this. Someone send me the next math: Sum($1+2+4+8+16+$..)= infinity Which I understand S=sum($1+2+4+8+16+$..) S=1+sum($2+4+8+16+$..) So this ...
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2answers
66 views

why is $\lim_{x\to -\infty} \frac{3x+7}{\sqrt{x^2}}$=-3?

Exercise taken from here: https://mooculus.osu.edu/textbook/mooculus.pdf (page 42, "Exercises for Section 2.2", exercise 4). Why is $\lim_{x\to -\infty} \frac{3x+7}{\sqrt{x^2}}$=-3*? I always find 3 ...
0
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2answers
66 views

Sum of alternate terms of Riemann Zeta function

If $\sum\limits_{n=1}^{\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}$ Then find the value of $\sum\limits_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}$ The book I took this problem from makes no mention of Riemann ...
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1answer
54 views

Are these “infinity” sequences true? [duplicate]

For $1\over 3$, you get $0.\overline3$, which is $0.33333...$. The threes go on forever. You can't ask "What happens if it ends in an eight?" because it simply doesn't end. For SSSSS..., what if it ...
2
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1answer
48 views

Number of equivalence classes of binary sequences which differ only by finitely many elements.

This question rose up when i was reading a problem the author used to argue against the axiom of choice. Consider the set of all (infinite) sequences of 0's and 1's. Q1) How many such sequences are ...
0
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3answers
158 views

Is the number of finite strings infinite?

I already asked this question on Stack Overflow and people kept voting me down and telling me it's "more of a maths question" so I will ask the question again: Assuming a finite alphabet, (eg: A,B), ...
0
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2answers
69 views

Calculating $ \lim_{n\to \infty} (1+\sin({1}/{n}))^{n}$ without L'Hopital or series expansions [duplicate]

I am trying to calculate the following limit, without using the L'Hopital rule or series expansions: lim (1+sin(1/n))^(n), n->infinity I now that it is the ...
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0answers
30 views

Logarithmic Series [duplicate]

I was doing a bit of math when I came across logarithmic series. I have no idea from where they come from. They seem so unrelated, that I have no intuition behind them at all. So, can anyone prove ...
0
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5answers
112 views

If $0{.}9\ldots$ is $1$, what does that make $0{.}3\ldots$?

So I recently learned that $0{.}9$ repeating is equal to $1$: $$ x = 0{.}9\ldots\\ 10x = 9{.}9\ldots\\ 9x = 10x - x = 9{.}9\ldots - 0{.}9\ldots = 9\\ x = 9x/9 = 9/9 = 1\\ x = 1 $$ Or a simpler ...
2
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4answers
172 views

Number of iterations to reach cosine's fixed point

I was messing around with my calculator the other day when I saw something interesting happen. Whenever I repetitively took the cosine of any number, it always ended up on a particular number ...
2
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2answers
81 views

'Smaller than infinity' notation

I've been coming across some papers (written in the 1960s - 1970s) that use the following peculiar statement: Let use denote by $H$ the space of all grid-functions $w_r$ for which: $$ ...
2
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2answers
63 views

Ice cream issue in Lem's 'Extraordinary Hotel'

Could you clarify the ice cream issue mentioned at the end of the story The Extraordinary Hotel (pages 189-190 here)?
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0answers
37 views

Is it true that the slope of a vertical line times the slope of a horizontal like don't equal $-1$, even though they're perpendicular?

I know that the slopes of two lines that are perpendicular have a value of $-1$ when multiplied because they're opposite reciprocals (e.g. $5$ and $-{1\over 5}$), but what if there's a horizontal and ...
0
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2answers
105 views

What is the one point compactification of the reals?

In several of my questions this theorem has come up. What is the one-point compactification of the reals? Does it have to do with limits and dividing by $0$? I vaguely remember something about a ...
4
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2answers
81 views

Is there any unreachable result?

I hope that this question is reasonable and make sense because I am not sure. Every theorem's proof is consisting of finite logical steps. Can a proof of the theorem require infinitely many ...