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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3answers
70 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: $$|\frac{(n+1)^{n+1}}{(n+1)!}\cdot ...
2
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1answer
57 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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0answers
33 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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5answers
94 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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0answers
24 views

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 ...
2
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5answers
91 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
28 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
32 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 ...
2
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2answers
49 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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2answers
44 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
48 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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2answers
24 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 ...
1
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1answer
35 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' ...
1
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2answers
44 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.
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2answers
36 views

Find $\lim_{x\to \infty} \sin\left(\frac 1x\right) $

Show that $$\lim_{x\to \infty} \sin\left(\frac 1x\right) = 0$$ I don't even really know where to start on this question. I know that the limit definition at infinity is: for all $\epsilon>0$, ...
3
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0answers
50 views

How does $\ln(x)$ blow up at $0$ and $\infty$.

In general: How do I figure out how fast a function blows up at a certain point or infinity? How fast does $\ln x$ blow up at $0$? Does it blow up as fast as $1/x$, $1/x^2$, or maybe faster than any ...
3
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1answer
52 views

Is the set of real numbers really uncountably infinite?

The proof that the set of real numbers is uncountably infinite is often concluded with a contradiction. In the following argument I use a similar proof by contradiction to show that the set of ...
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1answer
46 views

Perfectly Round Sphere on Perfectly Flat Floor

I know that in practice it may be practically impossible to create the following situation, but suppose I did place a perfectly round ball of radius r (not that I think radius r is relevant) on a ...
0
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0answers
61 views

Why is $y + 1$ infinite?

This is related to SO question : http://stackoverflow.com/questions/30150877/why-does-this-cause-ghci-to-hang but I'm having difficulty understanding why Haskell enters an infinite loop but since ...
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2answers
42 views

Evaluate the following improper integral.

$$ \int^{+\infty}_{-\infty} \frac{x\sin 4x}{x^2-4x+8}dx \, $$ My Thoughts: I know that I should start by changing the integral to: $$ \int^{+\infty}_{-\infty} ...
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0answers
34 views

Comparing ordered numerical sets by their density [closed]

Cantor's idea of comparing infinite sets by making one to one correspondences between their members seems to be very fruitful. Yet I wonder whether anybody ever suggested to compare ordered and ...
0
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1answer
38 views

Another flavour of Hilbert's Hotel

Many of you have probably heard about Hilbert's Hotel problem. Mr Hilbert owns a hotel with countably infinite amount of one-bed rooms. All the rooms are, of course, taken. A (finite or infinite) ...
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0answers
26 views

Laplace transform and “imaginary infinity”

I was recently studying Laplace transform for the first time, and I'd like to ask the following thing: there was an integral with limit of integration, something like that: a+j×infinity, j the ...
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3answers
38 views

Construct a non-linear function that shows that the intervals $[2,4]$ and $[10,22]$ have the same cardinality

Using something other than a linear function, show the intervals $[2,4]$ and $[10,22]$ have the same cardinality. I don't quite know where to start with this problem, or what key factor is necessary ...
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6answers
158 views

All numbers are undefined? [closed]

Notice the following set of equations $$\infty = \infty -x$$ $$x = \infty - \infty$$ $$x = \text{Not defined}$$ here $x$ can be any real number and hence all real numbers are undefined. This is ...
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4answers
63 views

Need assistance solving a limit question without applying l'hopital's rule

This is the question: $$\lim_{x \rightarrow-\infty} \frac{|2x+5|}{2x+5}$$ I know the answer is $-1$, but can someone go through the steps and explaining it to me?
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0answers
33 views

find the supremum

Hi I'm trying to figure out for which values of $w$ of $u(x,t)$ the absolute value of the supremum of $u(x,t)$ is infinity. The function $u(x,t)$ is the following. According to my calculation is ...
0
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1answer
36 views

If $A$, $B$, $C$ are any infinite sets then is $|A|=|B|$ and $|A|=|C|$ $\Longleftrightarrow |A|=|B\cup{}C|$?

Suppose we have three sets $A$, $B$, and $C$ that we know are infinite sets, but we do not know anything else about the cardinality of $A$, $B$, and $C$. Is $|A|=|B|$ and $|A|=|C|$ ...
1
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3answers
57 views

Uncountable infinity

The "number" of real numbers in $[0,1]$ is uncountably infinite, just as the "number" of real numbers in $[0,10]$ is uncountably infinite. However, my intuition would tell me the second interval has ...
0
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0answers
19 views

Compute a sum with finite and infinite elements

I would like to compute the following summation: $$ s = \sum_{i=1}^n a_i \, \Phi^{-1}(u_i) $$ where $\Phi^{-1}$ is the inverse of the standard Gaussian distribution function, $a_i$ are some real ...
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1answer
78 views

Question about infinity

That might be a silly question, but here goes: I see a lot of "big numbers" in physics, such as the size of the state space of all the particles in the visible Universe, and those numbers can be ...
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2answers
29 views

Infinite averages

If you wanted to find the average of infinite items, what would it be? Would an estimate of the first x averaged be a good estimate? Or would the value be nearly zero because you are dividing by ...
1
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0answers
35 views

How does one prove that two sequences are equal at infinity?

I came up with $e = \sum_{n=0}^\infty \frac 1 {n!}$ (see here) I am now trying to prove that this is equivalent to $\lim_{n\to \infty} {(1+\frac1 n)}^n$ In general, how would one go about such a ...
0
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1answer
44 views

Is the number of points on a plane larger than the number of points on a line?

The number of points on a line is uncountably infinite. The number of lines on a plane is uncountably infinite. It seems like it follows that there would be an uncountably infinite number of points on ...
0
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1answer
23 views

limits as $x\rightarrow\pm\infty$ of indeterminate forms $\frac{a^x+b^x}{c^x+d^x}$, where $a,b,c,d\in\mathbb{R}$

Good day sirs would you kindly help me to find the limit of $\frac{a^x+b^x}{c^x+d^x}$ as $x\rightarrow\pm\infty$, where $a$,$b$,$c$ and $d$ are real numbers? I already know how to use the L' ...
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0answers
33 views

How many different random numbers can you generate between zero and infinity?

My theory is that you can't generate any random number between zero and infinity. But thats kind of unexpected because you can generate 100 different random numbers between 0 and 100 and 1000 ...
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2answers
76 views

Bijection between $\mathbb{Z} \longmapsto \mathbb{R}$ [duplicate]

I recently learned of Cantor's diagonal argument, and was thinking about why there can't be a bijection between any infinite set of integers and any infinite set of real numbers. I understood the ...
1
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2answers
84 views

Every II-finite set is III-finite

I need some help proving that if a set $X$ is II-finite then it is III-finite, i.e. if every non-empty family of subsets of $X$ which is linearly ordered by inclusion has a maximal element under ...
1
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1answer
26 views

Asymptotic behaviour of sequences

Could anyone explain in details how these approximations as $n \to \infty$ are found? ($a$ is a positive real number) ${x_n} = \frac{1}{n}\left( {\frac{a}{3} - \frac{3}{2}} \right) + O\left( ...
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1answer
34 views

Is the derivative of a exponential function a^x always greater than the derivative of a polynomial x^n as x approaches infinity

with n and a being any constants > than 1. I have tried taking the $\lim\limits_{x \to \infty} a^x / x^n$, and l'hopitals is telling me than $x^n$ can always be reduced to 1 with multiple iterations, ...
0
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1answer
47 views

Problems with x/∞ [duplicate]

If $\dfrac {x} {\infty }=0,$ where $x$ is a finite number, than wouldn't $0\cdot \infty $ be equal to any number? Making this not work?
1
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2answers
34 views

Complex infinity ($1/0$) [duplicate]

I've learned that $$1/0$$ is postive and negative infinity, but if I ask wolfram mathematica to calculate $$1/0$$ it gives me: 'complex infinity' but how can we proof that that is true?
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2answers
67 views

Prove or disprove : if $a_n$ has a limit and $b_n$ doesn't have a limit then $a_n + b_n$ doesn't have a limit

I think it's wrong but I couldn't find an example that disproves this. If this is true I need to prove it and if it's wrong I have to give an example to disprove it.
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2answers
47 views

P vs NP and Countable vs Uncountable Decision Space

I have noticed that whenever the scope of a problem is pushed to infinity, problems in NP have an uncountably infinite decision space whereas problems in P seem to have a countably infinite decision ...
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1answer
46 views

Number of paths in a graph with infinite nodes

Does a graph with infinite nodes that is not fully connected have a countably infinite or a uncountably infinite number of paths originating from a single node? We are only concerned with paths that ...
1
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1answer
38 views

Doubt related to the extended real line and distance/metric

I am studying real analysis and I am being introduced to functions that take values on the extended real line, I have a fundamental doubt about this so I'll give an example to illustrate my confusion: ...
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4answers
109 views

Proof of 1+1+1+1+… = 0 [closed]

I have thought that 1+1+1+1+... is equal to -1/2. However I found this proof based on the assumption of 1+2+3+4+5+... = -1/12 and -1+(-2)+(-3)+(-4)(-5)+... = +1/12 1+1+1+1+1+1+... is equal to ...
0
votes
3answers
114 views

All sets of rational numbers are bigger than the set containing infinite integers - or are they?

Intro This started with me learning the different types of infinity. I like to call them types instead of sizes due to the fact, that infinite is defined by being endless or "not-finite" - meaning ...
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1answer
86 views

Extend ${\bigl(1+\frac1x\bigr)}^{{x}}$ to $\overline{\mathbb R}$

We can extend these functions to $\overline{\mathbb R}$ by taking limits says here. \begin{align} \mathrm e^{-\infty} &= 0 \\ \mathrm e^{+\infty} &= \infty \\ \ln{\left|0\right|} &= ...
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1answer
19 views

Limit at $\infty$ of a polynomial multiplied by a negative exponential

I am trying to show $\int_0^{\infty} x^2 e^{-2 x} dx = 1/4 $ Integration by parts gets the indefinite integral $$\int x^2 e^{-2 x} dx = \frac{-1}{4} e^{-2 x} (2 x^2+2 x+1)+constant$$ In order to ...