What is Infinite: Definition and 1000 Discussions

Homework Statement A function f(x) has the following series expansion: ##f(x)=\sum _{n=0}^\infty \frac{c_n x^n}{n!}##. Write down the function ##g(y)=\sum _{n=0}^\infty c_n y^n## under a closed form in function of f(x). Homework Equations Not sure at all. The Attempt at a Solution...

Alright, I'm back with yet another question... So the prof was explaining that the energy in an infinite cubical well is E((h2∏2)/2ma2))(nx2+ny2+nz2) Which is all well and good, and he gave us the example of: ψ1,2,1 = E = 6((h2∏2)/2ma2)) And with little explanation mixed it up once...

$\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$ where p is any fixed real number. If this was just the telescoping series or the p-series, this wouldn't be a problem.

Homework Statement Prove that if A,B, and C are nonempty sets such that A \subseteq B \subseteq C and |A|=|C|, then |A|=|B| The Attempt at a Solution Assume B \subset C and A \subset B (else A=B or B=C), and there must be a bijection f:A\rightarrowC...

Ok I understand the concept of infinite countability and that say the set of all rational #s is infinitely countable, but if I needed to represent the set how do I do that? S={xε rat. # : x= k , k ε a rational #}? that doesn't seem right. Also say I wanted to show a set of finite countable...

Homework Statement Determine the limit of the sequence: an = (1+(5/n))2n Homework Equations L'hopitals rule, or at least that's what I'm thinking. Otherwise, general formulas for determining the limit of a sequence. The Attempt at a Solution an = (1+(5/n))2n Considering the...

An electron is trapped in an infinite one-dimensional well of width 0.251nm. Initially the electron occupies the n=4 state. Suppose the electron jumps to the ground state with the accompanying emission of a photon. What is the energy of the photon? (Time independent) What I did was...

Homework Statement The problem is part of a review and we are only to determine if the series converges or diverges by any test, and state the test. ##\sum_{n=1}^\infty(\frac{k}{k+1})^k## My work so far I know that the root test gives an inconclusive answer and from there I moved...

Homework Statement Homework Equations -h^2/2m d^2F(x)/dx^2 = EF(x) The Attempt at a Solution i just need to a part. for E<0 i can find for 0<x<L side F(x) = ACos(Lx) + BSin(Lx) at the L<x side, F(x) = e^(Kx) where L^2= 2m(E+V)/h^2 K^2= -2mE/h^2 but i do not know what will i do. can...

Homework Statement Describe the potential inside and outside an infinite insulating sheet with uniform density ρ and thickness d, as a function of x (distance from the center of the sheet). zero potential has been set at its center. What is the potential on the surface of the sheet?Homework...

Hi I have attached my attempt of solving the infinite square well for Energy. The value I get is different from that of the book, also in the attachment, Kindly explain if my answer is correct given the fact that I proceeded step by step and used no tricks. Thank you.

Homework Statement To give some context, I'm trying to show that \mu(\bigcup^{\infty}_{k=1}A_{k})\leq \sum^{\infty}_{k=1}\mu(A_{k}) where μ is the Lebesgue measure and the A's are a countable set of Borel sets. Since the A's may not be disjoint, I'm trying to rewrite the left side of the...

Question says: \sum(cos(n*pi)/5^n) from 0 to infinity. Proved that it converges: ratio test goes to abs(cos(pi*(n+1))/5cos(pi*n)) with some basic algebra. As n goes to infinity, this approaches -1/5 (absolute value giving 1/5) since cos(pi*(n+1))/cos(pi*n) is always -1, excepting the...

Homework Statement A line charge starts at x = +x0 and extends to positive infinity. The linear charge density varies inversely with distance from the origin, λ(x)=(λ0*x0)/x derive the expression for the electric field at the origin, E0, due to this infinetly long line-charge (L→+∞)...

Homework Statement Show that \sum_{k=0}^{\infty} \sqrt[k]k-1 converges. Homework Equations Ratio, radix theorems, comparison with other sums... The Attempt at a Solution No idea whatsoever. Where does one begin in this case ? With other cases I'm quite confident.

I was asked to find the magnetic field of an infinite current sheet due to amperes law. Is my attempt to the solution correct ? The final answer is correct, but l am doubtful of how l got there.

Homework Statement An electron is trapped in an infinitely deep potential well 0.300nm in width. (a) If the electron is in its ground state, what is the probability of finding it within 0.100nm of the left-hand wall? (b) Repeat (a) for an electron in the 99th excited state (n=100). (c) Are...

Homework Statement Quantum mechanics is absolutely confusing me. A proton is confined in an infinite square well of length 10-5nm. Calculate the wavelength and energy associated with the photon that is emitted when the proton undergoes a transition from the first excited state (n=2) to the...

The relativistic mass formula is m=γm, and at the speed of light, relativistic mass is infinity. But, the Lorentz factor at the speed of light is 1/0, but this is undefined, so why do physicists call this "infinity"?

Homework Statement Consider an infinite, one-dimensional linear chain of dz2 orbitals separated at a distance a. Write an expression of the BLOCH FUNCTION that describes this chain. Homework Equations ψk=Ʃexp(ikna)χn The Attempt at a Solution I read this...

Prove that \(R^{\infty}\) is infinite dimensional.

Homework Statement Let's consider a distribution function f=f(t,x^i,E,p^i). Is it true that \mathop {\lim }\limits_{p \to\infty}p^{\alpha}f=0 \forall\alpha\in R ?Homework EquationsThe Attempt at a Solution I think so, not sure though. Thanks in advance!

→Homework Statement Integrate the improper integral (use correct notation). State whether it's converging or diverging. 10 ∫ 7/(x-9)^2 dx 8 Homework Equations b c ∫ f(x) dx= lim ∫ f(x) dx a c → d a The Attempt at a Solution...

I am having a difficult time seeing how \sum_{n=0}^{\infty} ((-1)^n + 1)x^n is equivalent to 2\sum_{n=0}^{\infty} x^{2n}

Homework Statement I need to show that both sin(x) and cos(x) are absolutely convergent. Here's my work so far, Theorem: ℯix = cos(x) + i*sin(x) (1) Proof: This...

Homework Statement An electron is trapped in an infinite square-well potential of width 0.5 nm. If the electron is initially in the n=4 state, what are the various photon energies that can be emitted as the electron jumps to the ground state?Homework Equations ΔE=13.6(1/nf2-1/ni2)...

I understand it is physically impossible for anything to move at an infinite speed simply because infinity can never be reached but... My understanding of physics is that as something interacts with the Higgs Field it is given mass and therefore requires more energy to move. However I'm also...

Hi, I'm looking into quantum computing, and if you could forgive my naivety I was wondering whether the superposition of a qubit could produce a one-step solution to an infinite salesman problem? The salesman problem is where a computer has to calculate the most efficient route for a man to...

I'm a little confused about the electron wavelength in an infinite potential well. It is my understanding that the maximum wavelength that the electron can achieve is 2 times the length of the potential well. As the eigenvalue increases, does the wavelength change? I believe that the...

Homework Statement For the single line charge, derive an expression for Electric Potential. Homework Equations V(r)=-\intE\bulletdr E for infinite line = \frac{\lambda}{2\pi r\epsilon} The Attempt at a Solution The integration is straightforward enough—my question is as to what the...

This might sound like a dumb question, but it's actually not too obvious to me. If we know that \lim_{n→∞}S_{n} = L , can we prove that \lim_{n→∞}S_{n-1} = L ? I'm actually using this as a lemma in one of my other proofs (the proof that the nth term of a convergent sum approaches 0), but...

According to my professor, there exist infinite dimensional vector spaces without a basis, and he asked us to find one. But isn't this impossible? The definition of a dimension is the number of elements in the basis of the vector space. So if the space is infinite-dimensional, then the basis...

From a fraction with infinite sum in denominator to partial fractions?? I am currently studying a course on Perturbation Methods and in particular an example considering the following integral \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\epsilon^2 + \sin^2 \theta}. There's a section of the...

This is a paradox that has been bothering me since I was taking algebra in high school. Let's say that I want to represent the distance between to objects. Given that numbers are infinite in both directions, by which I mean that there is no limit to how large, or small a number can be, there...

It's been said that the universe has no edge, it's expanding, it has no center and the big bang was the birth of energy, matters and space-time. I also often hear that it's been estimated the universe has approximately 200 billion galaxies or more or much more. Also the number of particles...

Hello, i would like to ask the following question that has been troubling me. lets say i have 1/3 = 0.33333333333(3) it may seem clear that 1/3 - 1/3 = 0, but when operating over the decimals this doesn't seem clear. How can i perform an operation over infinite digits and consider it as...

S = \frac{1}{2} + \frac{1}{4} + ... + (\frac{1}{2^n}) I noticed that this is a sum of a infinite series with the common ratio being 1/2, so using \frac{1}{1-1/2} I get S = 2, however with this question there is a hint saying multiply S by 2, which I did not use so I'm worrying if I done...

So I solved for series that I know is geometric, and I've been able to find the solution, but only because what was written in my notes. Personally it isn't sitting well with me because I don't see the relation to a simple geo series: Ʃ (wq)k = wq/(1- wq). Now if this is my series...

I am not getting anywhere with this problem. Prove the Schwarz's and the triangle inequalities for infinite sequences: If $$ \sum_{n = -\infty}^{\infty}|a_n|^2 < \infty\quad\text{and}\quad \sum_{n = -\infty}^{\infty}|b_n|^2 < \infty $$ then $\displaystyle\left(\sum_{n = -\infty}^{\infty}|a_n +...

Homework Statement The question is out of Hungerford's Algebra (Graduate Texts in Mathematics). Page 69,#7: Show that the group defined by generators a,b and relations a^2=e, b^3 = e is infinite and nonabelian. Homework Equations The Attempt at a Solution My professor gave...

This problem seems a little overwhelming at the point. I am not sure on where and how to start. Suppose that a uniform thermal gradient in the +x direction exists in a very large (i.e. effectively infinite) domain of conductivity $k_2$ such that the temperature field $u_{\infty}(r,\theta)$ can...

Homework Statement Show that infinitely many of the numbers 11, 101, 1001, 10001, 100001,... are composite Homework Equations The Attempt at a Solution So by inspecting these numbers, I notice that 11, 1001, 100001 are all divisible by 11. The numbers can be represented at 10^{n}+1 and...

in the infinite well with small potential shown in the attachment. I calculated the total energy by using the time independent Schrodinger equation and adding the correction energy to the equation of the slope k=(Vo/L)x. E=h^2/8mL^2 +∫ ψkψ dx ψ=√(2/L) sin⁡(∏/L x) when integrating ∫...

in the infinite well with small potential shown in the attachment. I calculated the total energy by using the time independent Schrodinger equation and adding the correction energy to the equation of the slope k=(Vo/L)x. E=h^2/8mL^2 +∫ ψkψ dx ψ=√(2/L) sin⁡(∏/L x) when integrating ∫...

Electric field near charged sheet is sigma/2E Which is independent of the distance from it.. However In case of point charge, as we go very close to it, magnitude of electric field tends to infinity.. But why doesn't this happen with charged sheet, i mean it can also be considered as...

Homework Statement I attached the solution to the problem. Homework Equations The Attempt at a Solution I can see that the infinite series diverges by looking at a few terms, but how would I find a general term for the infinite series, to evaluate it analytically?

Hi guys, this is more of a conceptual question, so I hope you guys can give me a detailed explanation if possible. Homework Statement Find the Electric Field inside a Charged Sphere (charge only on the surface) and between two Parallel Plates (oppositely charged) separated by some distance d...

I hope I'm not being redundant here. I would like to re-start discussion on how, according to quantum mechanics, the universe can exist. The question could go deeper and ask why, since the universe does in fact exist, did it become so complex, develop life on at least one planet, etc. How did...

Homework Statement Determine \sum_{k=0}^\infty \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}. Homework Equations ## (x)_m=x(x-1)(x-2)...(x-(m-1)) ##, integer ##m\geq0##. ## (x)_{-m}=\frac{1}{(x+1)(x+2)...(x+m)}##, integer ##m>0##. ## Δ((x)_m)=m(x)_{m-1}## \sum_{a\leq k<b}...

Homework Statement a_n = \frac{(2n -1)!}{(2n)^n} Homework Equations The Attempt at a Solution I am not exactly sure how to solve this problem.