What is Infinite: Definition and 1000 Discussions

Homework Statement Integrate √(25 - x^2) dx from 0 to 5 using an infinite Riemann Sum Homework Equations lim n→∞ Σ_(i=1)^n i = n(n+1)/2 lim n→∞ Σ_(i=1)^n i^2 = n(n+1)(2n+1)/6 The Attempt at a Solution Δx = (b - a)/n Δx = (5 - 0)/n Δx = 5/n f(x_i) = √(25 - [a + iΔx]^2) f(x_i) = √(25 - [0 +...

why do infinite loops occur? is it just because the condition is always true?

This isn't quite a calculus question, but it didn't seem right for any of the other mathematics forums, either. Does anybody if there is a closed form for the following infinite series: \sum_n x^{n^2} for 0 < x < 1

Please help me If I put a dynamo in a space ( no air ) ... and I gave it a movement ... to produce energy. ... and considered to delete any friction between its parts ... will it move forever producing infinite energy !?

> Q:The two charged strips in the following picture have width b , infinite height,and negligible thickness(in the direction perpendicular to the page).Their densities per unit area are \pm \sigma . >a)Find the field magnitude produced due to one of the strips from a distance a away from...

Homework Statement So this is part of a broader problem about the quantum harmonic oscillator, but there's one particular bit of mathematics I'm stuck on. We have the differential equation: y''(x) +(ε-x2) y = 0 And I'm told that we're to examine how y behaves as x tends towards...

How can speed of light be absolute yet that fact and relativity of sublight speed implies relativity of space and time intervals; if time time dilation is infinite at v=c, then time stands still only from that reference frame, but light takes a longer time relative to any other reference frame...

Homework Statement A square insulating sheet 75.0cm on a side is held horizontally. The sheet has 7.40nC of charge spread uniformly over its area. Calculate the electric field at a point 0.240mm above the center of the sheet. Homework Equations \Phi = 2EA E = \sigma / 2\epsilon The Attempt...

Homework Statement Infinitely long cylinder of radius R with uniform charge ρ. Calculate the electric potential at all points in space. Homework Equations V(a)-V(b)=-∫ba\vec{E}(\vec{r}')°dr'\hat{r} The Attempt at a Solution Generally potential is calculated with a reference...

What exactly prevents us from ruling out a uniform distribution on infinite sets? To be more precise, why are distributions and limits like \int_{-\infty}^{+\infty}dx\,\lim_{\sigma\to\infty}f_{\mu,\sigma}(x) = 1 \int_{-\infty}^{+\infty}dx\,\lim_{\Lambda\to\infty}\frac{1}{\Lambda} \chi_{[a,a+L]}...

I love having discussions with my friends about cosmology, physics and just the universe is general but can someone explain to me why ALL my friends tell me this way of thinking is just plain wrong. We were talking about whether or not we think the universe is infinite or not and I told them...

Homework Statement the effective capacitance across AB of the infinite ladder shown in the figure The Attempt at a Solution

So consider a function ##f(x)## which is continuous for all ##x## except on some finite interval, say ##[a, b]##. Imagine, for example, a function which goes to ##-\infty## from the left at ##x=a##, is undefined from a to b, and then "comes from" infinity at ##x=b## and is defined and continuous...

I was researching the inflationary model of the universe and came across the idea that the universe may be both infinite and expanding; and that there isn't a contradiction. As time goes by, the amount of matter in any given area will become less dense due to the metric expansion of space...

Is it possible to achieve infinite voltage?

Could someone walk me through how to maximize this 2-variable function wrt z? http://www.wolframalpha.com/input/?i=z+%3D100%2F%281%2B%28root+%28%28x-2%29%5E2+%2B+%28y-3%29%5E2%29%29%29+-+100%2F%281%2B%28root+%28%28x-2%29%5E2+%2B+%28y-3%29%5E2%29%29%5E2%29 I know the set of solutions will...

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the general UMP-based definition of external and internal direct products ... ... On page 63, Knapp defines...

Entropy gave us that everything can happen, just that the chances of it happening is very slim. If time is infinite, wouldn't everything that can happen will happen?

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding infinite direct sums and products and indexed families of sets ... ... On page 62, Knapp introduces direct...

Mathematicians have long held that infinite integers do NOT exist, but here is a very simple argument that shows that they do exist. A list of positive integers Z+ can be formed in a base-1 numeral system as... 1 11 111 1111 . . . 1111111... Since the set of integers is infinite...

Considering that speed of light is constant and finite, then why are the time dilatation and length contraction infinite to a frame of reference moving at the speed of light? We know that a moving frame of reference experiments time dilatation and length contraction from the point of view of a...

I've read that the Alcubierre Drive depends on the existence of negative mass, but I've seen that physicists say it could violate the conservation of energy. Their reasoning is that basically a negative mass and positive mass would interact in a perpetual motion sort of way that eternally...

As far as i understand the current big bang theory, it started as a extremely dense object, finite in size. But we still think (or well it is very accepted to belive) that the universe is infinite. I know inflation should be though as an expansion everywhere at the same time rather than the ball...

use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. $\sum_{n=1}^{\infty}\frac{1}{3^n+4^n}$

From Optics by Hecht He says "only a plane wave of infinite extent will propagate as a plane wave" What does it mean by " plane of infinite extent" in this context?

Homework Statement Find the potential difference between two oppositely charged, infinite cylinders of radii R whose axes lie at $$y =+\frac{d}{2}$$ and $$y = -\frac{d}{2}$$ They have surface charge densities of magnitude $$\sigma$$ Homework Equations The family favourite - Gauss' Law...

Homework Statement Given the following normalised time-independent wave function the question asks for the expectation value of the energy of the particle. The well has V(x)=0 for 0<x<a Homework Equations ψ( x ) = √(1/a) ( 1+2cos(∏x/a) )sin(∏x/a) The Attempt at a Solution I...

Hi. What are exactly the differences between the term "infinite" and "eternal"? Some said the term "infinite" means no end but have beginning/starting for example, Cantor's countable infinity. While Cantor's countable infinity have no end, but we can START begin counting it from 1 and so on...

Homework Statement Find the eigenfunctions of a particle in a infinite well and express the position operator in the basis of said functions.Homework Equations The Attempt at a Solution Tell me if I'm right so far (the |E> are the eigenkets) X_{ij}= \langle E_i \vert \hat{X} \vert E_j \rangle...

In multiverse cosmological models, what sort of infinities are usually used? For instance, in many models is it thought that the multiverse is a continuum - an unlistable (uncountable) infinite set comparable to the real numbers, and that it contains universes of Aleph_0 listable infinities...

Hi all, So I was recently set straight on the fact that bound state does *not* necessarily mean E<0 but rather is the statement that E<V(+/- infinity). So how do we apply this definition to the infinite square well where the potential at +/- infinity vanishes, and yet the bound states have...

Ok here's a potential I invented and am trying to solve: V = -Vo in -b<x<b and 0 in -a<x<-b , b<x<a where b<a and ∞ everywhere elseI solved it twice and I got the same nonsensical transcendental equation for the allowed energies: \frac{-k}{\sqrt{z_0 - k^2}} \frac{e^{2kb} +...

For the infinite square well in one-dimension the wavefunctions have the form Acos(kx) where k is the wavenumber which is proportional to momentum. Now due to H.U.P. if Δx is fixed as the infinite well size we can't know the exact momentum. I presume this is because the wavefunction exists as a...

I know some folks may get tired of questions about the finite/infinite scope of the universe. Sorry for that. But as you know, many concepts are hard to wrap one's head around. Let me make my question as clear as possible from the outset: -I am NOT asking whether the universe is infinite or...

Homework Statement Find the expectation value of the Energy the Old Fashioned way from example 2.2. Homework Equations ##\left< E \right> =\frac { 480\hbar ^{ 2 } }{ \pi ^{ 4 }ma^{ 2 } } \sum _{ odds }^{ \infty }{ \frac { 1 }{ { n }^{ 4 } } } ## The Attempt at a Solution Never...

Homework Statement On a long dielectric line a charge with density ##10^{-3}## is applied one half with positive charge and the other half with negative charge. Perpendicular to the first line and 5 cm away from it we have another line with the same charge density and also half of it is...

For every infinite value, there are an infinity of values less than it that are finite (since infinity minus one equals infinity). So wouldn't a huge but finite universe with very slight, undetectable curvature be infinitely more probable than an infinite flat universe? If that is so, let's...

Infinite Products This weeks challenge is a short one: Please list any sources that you have used to solve this question. Using google or other search engines is forbidden. Wikipedia is allowed but not its search engine. Wolfram alpha and other mathematical software are not allowed...

Can the difference between the two numbers (one) be divided infinitely? If so, are discrete counted numbers actually separated by an infinite "space"? Does this relationship between the discrete and the continuous (infinite) lie at the heart of reality? * I am not a scientist, as you...

Homework Statement Recognize the series $$3-3^3/3!+3^5/5!-3^7/7!$$ is a taylor series evaluated at a particular value of x. Find the sumHomework Equations Sum of Infinite series = ##a/1-x## The Attempt at a Solution So, I can't figure out what i would us as the ratio (the thing you multiply...

Hello, I've been reviewing some calculus material lately and I just have a couple questions: 1) I've seen infinite series shown graphically as a collection of rectangular elements under a curve representing an approximation of the area under the curve. But the outputs of the infinite...

How can we prove $$\displaystyle \tan^{-1}\left(\frac{4}{7}\right)+\tan^{-1}\left(\frac{4}{19}\right)+\tan^{-1}\left(\frac{4}{39}\right)+\tan^{-1}\left(\frac{4}{67}\right)+...\infty = \frac{\pi}{4}+\cot^{-1}(3)$$ My Trial: First we will calculate $\bf{n^{th}}$ terms of Given Series...

I ask this because it seems that there is no distinction between 0 and an infinitesimal. Similarly, it also seems that an infinite number of one dimensional lines can equal R2, and the same seems to go for R2 to R3, and R3 to R4. I only know basic calculus, so I am probably generalizing the...

In certain forms - including the logarithmic - a number of the trigonometric and hyperbolic functions can be used to sum series having Riemann Zeta and Dirichlet Beta functions (in the general series term). In this tutorial, we explore some of these connections, and present a variety of Zeta and...

dear all, let's consider a length L and divide it into N number of small segments uniformly. Then the length of every segment should be L/N. then we add these segments up, which is L=\sum\frac{L}{N} then we take the limit N→\infty at both sides, this means...

Homework Statement Prove that: [n+1 / n^2 + (n+1)^2 / n^3 + ... + (n+1)^n / n ^ (n+1) -> e-1 Homework Equations I have been trying it for couple of days. Tried to work the terms, natural log it all, use the byniomial theory but i can´t get to the right answer. The Attempt at a...

Homework Statement Sketch the difference of probability distributions at the two times. Does the energy change with time? The potential well suddenly disappears, what is the form of the wavefunction? Homework Equations The Attempt at a Solution Part (a) At t = 0, the probability...

What happens to ψ in a infinite potential well when the width is suddenly reduced to half its previous value ? Will this instantly adjust ψ to the new size of the well or will it take some time to confine itself in this new well ? And is there a possibility of quantum tunneling here?

Hey! :o I have to solve the following problem: $$u_t=u_{xx}, x \in \mathbb{R}, t>0$$ $$u(x,0)=f(x)=H(x)=\left\{\begin{matrix} 1, x>0\\ 0, x<0 \end{matrix}\right.$$ I have done the following: We use the method separation of variables, $u(x,t)=X(x)T(t)$. I have found that the eigenfunctions...

Hi! Here is my task: Calculate Iout if all BJT's have identical characteristics with β→∞, work on same temperature in forward active mode. Use BJT exponential transfer characteristic. Since β→∞, IB for every BJT should be zero amps and IC=IE. Equation for BJT exponential transfer...