Infinite Definition and 1000 Threads

1. Homework Statement . Prove that an infinite chain contains an a chain isomorphic (with the order) to N (the natural numbers) or to -N (negative integers). 3. The Attempt at a Solution . I think I know how to solve the problem but I have problems to write a formal proof. I want to...

1. Homework Statement . Let A and B be disjoint sets, A infinite and B enumerable. Prove that there exists a bijection between AuB and A. 3. The Attempt at a Solution . I have an idea of how to prove this statement, but I got stuck in the middle, so here is what I've done: There are just...

What do you think of this argument? Lets suppose an infinite and eternal universe that is homogenous on a large scale. Since there is no privileged inertial reference frame, regardless of the chosen inertial reference frame, the observed statistical velocity distribution of matter is nearly...

I imagine most everyone here's familiar with the proof that there's an infinite number of primes: If there were a largest prime you could take the product of all prime factors add (or take away) 1 and get another large prime (a contradiction) So what if you search for larger primes this...

Find \sum_{n=0}^\infty \frac{\cos(nx)}{2^n}.

Homework Statement The sum of the infinite terms of the series \text{arccot}\left(1^2+\frac{3}{4}\right)+\text{arccot}\left(2^2+\frac{3}{4}\right)+\text{arccot}\left(3^2+\frac{3}{4}\right)+... is equal to A)arctan(1) B)arctan(2) C)arctan(3) D)arctan(4) Ans: B Homework Equations The Attempt at...

Homework Statement A cubical box whose sides are length L contains eight electrons. As a multiple of $$\frac{h^2}{2mL^2}$$ what is the energy of the ground state of the eight electrons? Assume the electrons do not interact with each other but do not neglect spin. Homework Equations...

Hi All, I would like to know why in the infinite well problem, after having solved the time independent SE, we are not supposed to equal to zero the x derivative of the spatial part of the wave function at -L and L (2L being the total width). We only have to make it zero at the boundary...

According to a result of Paul Cohen in a mathematical model without the axiom of choice there exists an infinite set of real numbers without a countable subset. The proof that every infinite set has a countable subset (http://www.proofwiki.org/wiki/Infinite_Set_has_Countably_Infinite_Subset) is...

hello ... I propose this exercise for you to solve on various methods ...\[\lim_{n \to{+}\infty}{\frac{1}{n}\sum_{i=1}^n({1+\frac{i}{n}}})^{-2}\]thanks att jefferson alexander vitola(Smile)

Hello everybody, I'm trying to understand some steps in the evolution of calculus, and in a .pdf found in the internet I read the document: http://www.ugr.es/~mmartins/old_web/Docencia/Old/Docencia-Matematicas/Historia_de_la_matematica/clase_3-web.pdf , in pags. 14-15. I want to solve the to...

Homework Statement Homework Equations Stationary Schrodinger equation. The Attempt at a Solution 1st I draw the image of the well, so we can talk better - otherwise this makes no sense as it looks like a complex homework. In the image ##W_p## marks the potential energy but never mind i ll...

Hello! Unfortunately, I have not spent as much time as I should have on limits, or sequences, or their properties. In trying to work on a number theory math proof I have come across the following: I have an infinite sequence of numbers, all between 0 and 1 inclusive. I know that the limit of...

Homework Statement Particle is in an infinite square well of width ##L## on an interval ##-L/2<x<L/2##. The wavefunction which describes the state of this particle is of form: $$\psi = A_0\psi_0(x) + A_1\psi_1(x)$$ where ##A_1=1/2## and where ##\psi_0## and ##\psi_1## are ground and first...

1) Show that for $n >1$, $\displaystyle \prod_{k=1}^{\infty} \left( 1- \frac{z^{n}}{k^{n}} \right) = \prod_{k=0}^{n-1} \frac{1}{\Gamma\left[ 1-\exp (2 \pi i k/n) z\right]}$.2) Use the above formula to show that $ \displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) =...

Homework Statement Let A be a set, prove that the following statements are equivalent: 1) A is infinite 2) For every x in A, there exists a bijective function f from A to A\{x}. 3) For every {x1,...,xn} in A, there exists a bijective function from A to A\{x1,...xn} Relevant...

Time should have not have happened in infinite gravity of the Big bang Singularity. Einsteins General relativity suggest that time could not exist in a gravity field that was infinite. Thus, my question is how did the universe emerge from the Big Bang moment , if time and space did not yet...

In an earlier post i was shown how to represent an integral as an infinite sum. So why is the anti derivative a summation by definition? For example, the derivative dy/dx is found by f(x+h)-f(x)/h.

Two infinite-plane non-conducting, thin sheets of uniform surface charge p1 = 12.30 uC/m2 and p2 = -3.30 uC/m2) are parallel to each other and d = 0.615 m apart. What is the electric field between the sheets? (Note: the field is positive if it is parallel to the vector x). Hi, I've tried this...

Recently, an AT&T commercial has been running on TV where a moderator asks some children about the largest number they could think of. At the end, one kid replies “∞ times ∞”, which of course is simply ∞2. Natually, one can instantly think of a larger number: ∞∞. But then, that got me...

Hi everyone! Recently, I've been trying to understand how the error function pertains to solving for concentration in a non-steady state case (with a constant diffusivity D), but I've been having some trouble with the initial assumptions. The source I am currently using (Crank's The...

It seems to me that the question as to whether the universe is infinite or not carries the same validity as the question as to electron, quarks, etc. being infinitesimal or otherwise stated being modeled as point particles. It seems to me that these two quandaries are linked and perhaps can...

Homework Statement I'm having a bit of trouble following my textbook, I was under the impression ψ(x) = e^i(kx) = Cos(kx) + iSin(kx) but in my textbook they write the general solution to this equation as ψ(x) = ASin(kx) + BCos(kx). How come they wrote the sin part as not imaginary? isn't this...

Homework Statement proof this limit: \lim_{x\rightarrow 1^+}\frac{1}{(x-1)(x-2)}=-∞ Homework Equations The Attempt at a Solution So for every N < 0, I need to find a \delta > 0 such that 0 < x - 1 < \delta \Rightarrow \frac{1}{(x-1)(x-2)} < N Assuming 0 < x - 1 < 1, I get...

How to find the value of an infinite series. for e.g.Ʃ_{n=1}^{\infty} (β^{n-1}y^{R^{n}}e^{A(1-R^{2n})}) where β<1, R<1, y>1, and A>0? Note that this series is covergent by Ratio test. I already have the numerical solution of the above. However, I am interested in analytical solution...

Is it true that the universe can be infinitely large but still expand, so that at every particular moment in time the universe is infinitely large, but then becomes 100 light years larger for every second that time goes on for example? For example \infty+100=\infty I know infinity isn't a...

Hi, Homework Statement Suppose I have an infinite cylinder with radius R, axis along the z axis and constant magnetization M\hat{z}. I wish to find the magnetic field everywhere. (This is not a HW question per se, yet thought I might get some comments on my attempt at solving it...

Let f(x) and g(x) be non-piecewise defined functions that are defined for all real numbers. Furthermore, let f(x) and g(x) be continuous and differentiable at all points. Are there two functions f(x) and g(x) such that f(x)=g(x) for all points over some interval (a,b], and f(x)≠g(x) for all...

If an object is infinitely dense, does this simply mean that there is no empty space within the object? I'm hung up on the fact that you can't possibly get denser than infinite density; what is stopping a black hole from getting even denser? What happens to atoms once they're under such intense...

I was looking at this topic: http://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only-want-boys-closed And the top answer uses the fact that the sum from 1 to infinity of 1/(x2^x) is log 2. Why is this true? Thanks in advance.

I am just starting out in self-study for quantum theory, so forgive me if my question seems elementary or completely misguided. In quantum mechanics, every wave function ψ can be decomposed into a linear combination of basis functions in the following manner: \Psi = \Sigma{c_{n}\Psi_{n}}...

Problem: Prove that if an element is in the union of two infinite sets then it is not necessarily in their intersection: Proof: Have I solved it correctly?

So I was trying to see if \Sigmaln(\frac{n}{n+1}) diverges or converges. To see this I started writing out [ln(1) - ln(2)] + [ln(2) - ln(3)] + [ln(4) - ln(5)] ... I noticed that after ln(1) everything must cancel out so I reasoned that the series must converge on ln(1) which equals ZERO...

If you unite infinitely many open sets you still get an open set whilst the same is not necessarily true for a closed set. Can someone try to explain what property of a union of open sets it is, that assures that an infinite union is still open (and what property is the closed sets missing?)

How does one show that the set of polynomials is infinite-dimensional? Does one begin by assuming that a finite basis for it exists, and then reaching a contradiction? Could someone check the following proof for me, which I just wrote up ? We prove that V, the set of all polynomials over a...

Homework Statement There is a large\infinite amount of balls in a basket to pick from. Each ball in the basket is with the same probability (33.33...%) either black, white or gray. No other colors exist. You first pick 4 balls out of the basket. Then you pick 2 more balls out of...

Two infinite sheets of current flow parallel to the y-z plane as shown. The sheets are equally spaced from the origin by xo = 7.5 cm. Each sheet consists of an infinite array of wires with a density n = 19 wires/cm. Each wire in the left sheet carries a current I1 = 3.5 A in the negative...

The set of all functions is larger than 2^{\aleph_0} . So let's say I wanted to average over all functions over some given region. that was larger than 2^{\aleph_0} how would I do that.

is it also possible to transform any these kinds summation to any product notation: 1. infinite - convergent 2. infinite - divergent 3. finite (but preserves the "description" of the sequence) For example, I could describe the number 6, from the summation of i from i=0 until 3. Could I...

Homework Statement I think this is a very easy problem, I will try to show you guys what I tried to come up with: A particle is in an infinite potential box and is described in a certain moment of the normalized wavefunction ##\psi(x)=\sqrt{\frac{8}{3a}}sin^2(\frac{\pi x}{a})## for (0<x<a)...

I have been thinking about the idea that the universe is infinite in size and have wondered if it is not also possible that the universe is infinite in time as well - that is to say, the universe has always been here and always will be here, that it doesn't have a beginning or an end. This idea...

Homework Statement Consider a particle in 1D confined in an infinite square well of width a: $$ V(x) = \begin{cases} 0, & \text{if } 0 \le x \le a \\ \infty, & \text{otherwise} \end{cases} $$ The particle has mass m and at t=0 it is prepared in the state: $$ \Psi (x,t=0) = \begin{cases} A...

Hii All, Can anyone give me a hint to evaluate \sum_{n=1}^{\infty}\frac{a^{n}}{n^{1-m}}; Here 0<m,\,a<1. Please note that the summation converges and < \frac{a}{1-a}. A tighter upper bound can be achieved as 1+\int_{1}^{\infty}\frac{a^{x}}{x^{1-m}}dx. Is there any way to get the exact...

Homework Statement Homework Equations Countable Infinite is defined if X is infinite and X is isomorphic to the Natural Numbers. The Attempt at a Solution Now I assume that XUY is isomorphic to the Natural Numbers. So X ∪ Y ≅ N . Now here's where I get confused. I am...

Given S, an Infinite Series Summation, find \frac{1728}{485}S S=1^2+\frac{3^2}{5^2}+\frac{5^2}{5^4}+\frac{7^2}{5^6}+... I found out the formula for (r+1)th term of the series, hence making the series asS=1+\sum_{r=1}^{\infty}\frac{(2r+1)^2}{(5^r)^2} Now I have a hard time guessing what to do...

If we think of a functional as a function of the infinite number of taylor coefficients of the variable function, aren't they then just normal functions, a map between a set of reals to another set of reals.

Find the series sum ln2/2 – ln3/3 + ln4/4 – ln5/5 + ….

Hello there, I posted the very same question before, nonetheless received no answers since -i presume- new posts arose and mine went on to the back of the data registry. I know many of you are math and physics experts, that's why I want you to please help me find out the integral of a product...

Hi everyone ;) I have a challenging problem which I would like to share with you. Prove that \[\frac{1}{2^2}+ \frac{1}{3^2} \left(1+\frac{1}{2} \right)^2+\frac{1}{4^2} \left( 1+\frac{1}{2} +\frac{1}{3}\right)^2 + \frac{1}{5^2} \left( 1+\frac{1}{2} +\frac{1}{3}+\frac{1}{4}\right)^2 +\cdots=...

Around time 33:40 min in this video, Prof. Paul Steinhardt says the following about gravity: This argument is often brought forward, as in this case here, when inflationary cosmology is explained. The energy of the inflating universe comes from gravity or energy of the inflating universe is...