What is Infinite: Definition and 1000 Discussions

How to prove that \[ \sum_{i=1}^{\infty}\frac{1}{2^{3i}}\left(\csc^{2}\left(\frac{\pi x}{2^{i}}\right)+1\right)\sec^{2}\left(\frac{\pi x}{2^{i}}\right)\sin^{2}\left(\pi x\right)=1 \] for all \( x\in\mathbb{R} \). Using graph, we can see that the value of this series is 1 for all values of x...

how could something infinite expand? expandig mean geating bigger but if its already infinite how could it get bigger

I am trying to numerically solve (with Mathematica) a relativistic version of infinite square well with an oscillating wall using Klein-Gordon equation. Firstly, I transform my spatial coordinate ## x \to y = \frac{x}{L[t]} ## to make the wall look static (this transformation is used a lot in...

Context: What I don't understand is why the little current that flows through the converted voltmeter can affect the measurement when the resistance of the resistor added to the ammeter is not infinite. Can someone please explain this to me? Thanks.

A space is infinite dimensional when its basis is infinite. But how can I ensure that the basis of the space of all sequences whose limit is zero is infinite? (After solving that, I would like to have a hint on this very similar problem which says: let V be a Linear space of all continuous...

ok so I've been curious about this theory for a while but just haven't started digging into it until tonight : if energy creates gravity no matter how small, couldn't the universe be infinite and coincide with Newtons second law of thermodynamics; if when the heat death of the universe happens...

As shown in figure below, the electric field E will be normal to the cylinder's cross sectional A even for distant points since the charge is distributed evenly all over the charged surface and also the surface is very large resulting in a symmetry. So the derived formula should also apply to...

Summary:: How to know which one is bigger when n goes to infinity? $$ \sum_{n=1}^\infty \frac {1} {\sqrt {n}(\sqrt {n+1}+\sqrt {n-1})} $$ And: $$ \sum_{n=1}^\infty \frac {1} {\sqrt {n}(\sqrt {n}+\sqrt {n})} $$ I thought at first that the second one is bigger, although, I came to realize, to my...

Hello everyone, I have a problem with bounds states of the 1D Weyl equation. I want to solve the Dirac equation ##−i\hbar \partial _x\Psi+m(x)\sigma _z \Psi=E\Psi## with the mass ##m(x)=0,0<x<a##, ##m(x)=\infty,x<0,x>a##. ##\Psi=(\Psi_1,\Psi_2)^T## is a two component spinor. Outside the well...

Using Gauss's Law By using a symmetry argument, we expect the magnitude of the electric field to be constant on planes parallel to the non-conducting plane. We need to choose a Gaussian surface. A straightforward one is a cylinder, ie a "Gaussian pillbox". The charge enclosed is...

At the risk of waiting hours on simulations of a sensor, I was wondering if I could use infinite element domain on COMSOL to simplify it. The first image consists of what I would like to simulate but found out that the simulation time is a huge factor as I have a lot to simulations to conduct...

Summary:: Show that every infinite Turning-recognizable language has an infinite decidable subset Sipser's Theory of Computation, third edition, chapter three contains and exercise that asks us to demonstrate this. I don't know how to do this; I have certain ideas. We could modify the...

Suppose we measure some speed or energy of something with a suitable device or instrument. Now suppose the quantity that is being measured exceeds the capabalities of the measuring device either from above or below. How can we know if this quantity is indeed finite and not infinite?

Cyclic models for reference. I will take simple Big Bounce as an example of what I have in mind. In Big Bounce there periods of expansion and periods of contraction which result in a never-ending series of Big Bangs. However if Universe is infinite in extent this would require infinite amount...

In the standard framework of ideas about cosmology, is it possible to have a universe that is infinite in extent?

There is an entry in Wikipedia at this link: https://en.wikipedia.org/wiki/Pythagorean_triple Under elementary properties of primitive Pythagorean triples, general properties,sixth bullet from the bottom of this section, there is this generalized Diophantine equation: x^2p + y^2p = z^2 Where: p...

In particle terms I'm thinking this could be when the signal's photon density drops low enough to be indistinguishable from the vacuum energy. In wave terms, I'm thinking maybe either the amplitude or frequency might drop below any quantum threshold needed to budge an electron shell in any atom...

Hello folks, I want to simulate a 2D heat transfer process in the subsurface on a region which is infinite on the r-direction. So, as you know, the very basic way to model this is to draw a geometry that is very long in the r direction. I have done this, and the results that I obtain is correct...

If the maximum separation between two points on an infinite line is finite, then what is its value? So the maximum separation is infinite. Does this mean two points on an infinite line can be separated by an infinite distance? Why, why not?

I don't need equations, I would just like to pose a question which contradicts the above statement (I know I am wrong btw, I want to see where I am going wrong).My understanding of space (not near any gravity and therefore no spacetime curvature) is that a body in motion will continue to move at...

Summary:: Ideas concerning an ocean without limit. I asked this question in a philosophy forum and got quite a bit of feedback but it didn't quite answer my initial question. If there was an infinite ocean and I scooped a cup of water out of it, was anything actually taken/lost from the ocean...

So I am a layman in physics, I admit I am trying to grasp big ideas piecemeal via articles, wikipedia and YouTube. I don't pretend to be educated in this regard but I am curious and willing to learn! The idea of the multiverse intrigues me. Sidestepping for a second the fact that the idea has...

Using the equation En = (h2*n2 ) / (8*m*L2), I got that E1 = 0.06017eV but the answer is not correct.

After watching this video: which explains why the wavefunction in an infinite square well is flattened, I tried running the calculation in both, what seems, the more more traditional way of using sin and by the method of, what seems to be, adding the wavefunction and its complex conjugate...

General state of the infinite potential well is that ##L^2[0,L]##, where ##L## is well width, or ##C^{\infty}_0(\mathbb{R})##?

How do I find a coefficent of x^9 in a power series like this:

I have the following definition: $$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$ From this, how can I get the definition of $$\lim_{x\to p^-}=-\infty? $$

I have the following definition: $$\lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$ From this, how can I get the definition of $$\lim_{x\to p^-}=-\infty? $$

I have noticed that many PF participants seem to favor a cosmological model that our universe is infinite (and flat), but some (like myself) favor a finite universe. I have become curious about the subjective probability distribution, so I am hoping that many PF participants will post their...

I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?

Let us consider the infinite products ## p_{n}\,=\, 2\cdot 2\cdot 2 \cdot 2 \cdots 2 \,=\, 2^n## with ##n=1,\ldots ## . Clearly ##p_{n}\rightarrow +\infty## as ##n\rightarrow +\infty##. But if I put the infinity case ## 2\cdot 2\cdot 2 \cdot 2 \cdots \,=\, x## I have ##2\cdot x =x ## so...

Will translation parallel to x-axis work ? Else please suggest the symmetry? And does symmetry here refer to the symmetry of the sheet which causes the symmetry of the field or something else? Please be kind to help.

I took the w derivative of the wave function and got the following. Also w is a function of time, I just didn't notate it for brevity: $$-\frac{\sqrt{2}n\pi x}{w^{3/2}}cos(\frac{n\pi}{w}x) - \frac{1}{\sqrt{2w^3}}sin^2(\frac{n\pi}{w}x)$$ Then I multiplied the complex conjugate of the wave...

If the universe is either infinite, or it repeats, then I would assume that it would be possible for my atoms to come together again at some incomprehensibly long amount of time after my death. If this were the case, would my consciousness that I am currently experiencing now ever exist again...

Regarding finding centers of mass of infinite figures, how one can show that $$ \int_{-\infty}^\infty \left(\frac1{x^2}-\cos \frac1x\right)dx=\pi $$ for instance, and other similar integrals, like the following? $$ \int_0^\infty (x^2-\frac6{x^4})dx=0 $$

here is the question, don't mind about point (a) and (b) because i have solved them already...the main problem is the question on point (c) : so far, what i have done is : H = 2.7*0.1-(1.4*0.15+1.3*0.25) = -0.265 az A/m which is the wrong answer compared to the solution provided from the...

(a) I guess I should find ##C_n## by normalizing ##\psi_n##. $$∫_{∞}^∞|C_nψn(x)|^2 dx=C_n^2 \frac{2}{a}∫_0^a sin^2(\frac{πnx}{a})dx=1$$ $$C_n^2 \frac{2}{a}[\frac{a}{2}−\frac{a}{4πn}sin(\frac{2πna}{a})]=1⇒C_n=1$$ (b) $$Hψ_n(x)=\frac{-ħ^2}{2m}\frac{\partial^2}{\partial...

Off the back of a recently closed thread where there was some discussion about the gravitational field of an infinite flat slab, I decided to have a play at investigating that. I've found a few interesting things. It's fairly straightforward to solve for this situation. You use Cartesian-esque...

$\tiny{45.4.T40}$ Suppose that the coefficient matrix of a consistent system of linear equations has two columns that are identical. Prove that the system has infinitely many solutions. Refer to the DTSLS Diagram \item using augmented matrix A for example with c1 and c3 identical $\left[...

Over the years the following has continued to be my biggest question in Cosmology. In the past couple of years I wondered if we have got any closer to understanding whether our space is infinite or infinitesimal? (By infinitesimal I mean that there is no lower limit to the minimum separation of...

I am currently trying to compute the Green's function matrix of an infinite lattice with a periodicity in 1 dimension in the tight binding model. I have matrix ##V## that describes the hopping of electrons within each unit cell, and a matrix ##W## that describes the hopping between unit cells...

$$B = \frac {\mu_0 I}{2 \pi r} $$ By Right-hand Grip Rule, the direction of the magnetic field by wire in y-axis is into the paper (z) while the direction of the magnetic field by wire in X-axis is upwards (+i) The answer state the Magnetic field is in the (i - y) direction though. Next...

Sometimes I would like to transform an integral ##F(x) = \int_{a}^{x}f(s)ds## into an infinite integral of the form ##F(x) = \int_{0}^{\infty}f(g(u),x)du##. Is there some kind of change of variables that can guarantee this conversion on the boundaries and still give me a function of ##x##, at...

Obviously a particle inside an ISW of width L cannot have arbitrarily precise momentum because ΔP ≥ ℏ/2ΔX ≥ ℏ/2L. Therefore you cannot have a particle with arbitrarily low momentum, since that would require ΔP be arbitrarily small. I need to show that a photon inside an ISW cannot have...

The answer is that the charge density would be -σ, I cannot for the life of me understand why would that be the case. Of course it makes sense but I can't convince myself that it would be the only possible answer. I have tried to apply Gauss law a few times, but it doesn't yield anything.

Here are the results from the python code: Odd results: Even results: I tried to solve for energy using the equation: I substituted the value for a as 4, as in the code the limit goes from -a to a, rather then 0 to a, and hence in the code a = 2, but for the equation it would equal to 4...

I'm trying to get from the formula in the top to the formula in the bottom (See image: Series). My approach was to complexify the sine term and then use the fact that (see image: Series 1) for the infinite sum of 1/ne^-n. Then use the identity (see image: Series 2). Any other ideas?

what I've done so far? -i've determined the vector between the point (4, 0, 0) and the point P. (4, 6, 8) - (4, 0, 0) (0, 6, 8) -The norm of this vector is the radial distance of the line to point P (the value of “ρ” in the formula) √(0^2 + 6^2 + 8^2) = 10 -> ρ = 10 -and its unit vector is...

At first I take the uniformly distributed charge and then divide it by the area of the carpet to get the surface charge density σ -10E-6 C / 8m^2 = σ = -1.25E-6C/m^2 Then I divide the surface charge density by 2e0 to get the electric field strength caused by the infinite plane...

The book is Calculus: Basic Concepts for High School on the first page you are given the following sequence: 1, -1, 1/3, -1/3, 1/5, -1/5, 1/7, -1/7, ... several pages later the rule is given: in the second rule, for the first term in the sequence, the coefficient of one of the terms is 1/0...