Expansion Definition and 1000 Threads

Homework Statement Find the coefficient of x8 in the expansion of (x2-3)7 Homework Equations This one: (n n-j) an-jxj The Attempt at a Solution Hi! Well, I know that x= x2 a= -3 n= 7 and apparently, j=8. This is what confuses me. n-j is a negative number.. how would...

Homework Statement A brass bar and an aluminium bar are each attached opposite each other to immovable walls. (There is a diagram, but I think the idea is clear enough). There is a gap between the two bars of 1.3x10-3m at 28°C. At what temperature will the gap be closed? coefficient of...

Calculation of $57^{th}$ digit in the decimal expansion of $\displaystyle \frac{1}{57}$

Hello, according to my textbook, the Taylor expansion of first order of a scalar function f(t) having continuous 2nd order derivative is supposed be: f(t) = f(0) + f'(0)t + \frac{1}{2}f''(t^*)t^2 for some t^* such that 0\leq t^* \leq 1 Quite frankly, I have never seen such a formulation...

Homework Statement Find the Laurent expansion for \frac{1}{z^2-1} in the annulus 1 < |z-2| < 3 The Attempt at a Solution I've gotten to the last parts but getting stuck there. First I expanded the denominator and did a partial fraction decomposition and arrived at...

From my own(simplistic)perspective,dark energy is expanding the universe by creating further spacetime at a velocity faster than the speed of light,if gravitational waves propagate at the speed of light how is it possible that M31 and the milky way are still bound by gravity when the fabric of...

Hey guys. So I need to know how to Binomial expand the following function \frac{1}{(1-x^{2})}. I need this because I have to work out \prod^{∞}_{i=1}\frac{1}{(1-x^{i})} for i up to 6. But I have to do it with Binomial expansion. If i can learn how to do \frac{1}{(1-x^{2})} then the rest...

Hello, I'm fiddling with Wolfram Alpha and I can't find a definition of what do they mean by the "Series expansion of the integral at x -> inf". In particular, I have two divergent integrals and I am wondering whether their ratio is some finite number. Here it is: \left[\int_0^{\infty}...

Homework Statement Consider a series of three charges arranged in a line along the z-axis, charges +Q at z = D and charge -2Q at z = 0. (a) Find the electrostatic potential at a point P in the x, y-plane at a distance r from the center of the quadrupole. (b) Assume r >> D. Find the...

Homework Statement A rigid thick walled insulating chamber containing a gas at a high pressure ##P_i## is connected to a large insulating empty gas holder where the pressure is held constant at ##P_A## with a piston. A small valve between the two chambers is opened and the gas flows...

Homework Statement We know that sin(2⋅x)=2⋅sin(x)⋅cos(x)and cos(2x)=cos^2(x)−sin^2(x). Use the appropriate formula to expand cos(8x) in terms of one of these two formulas. Homework Equations sin(2⋅x)=2⋅sin(x)⋅cos(x) cos(2x)=cos^2(x)−sin^2(x) The Attempt at a Solution I am not...

I'm new to the forum, so please be kind. I was reading through my pchem textbook, and I noticed something. We're given the equation: ΔU = q + w For an adiabatic expansion, we're told that q = 0. Fair enough, no heat transfer. But when there is a constant T and change in V, my book...

Frank Wilczek and Larry Krauss point out (what is perhaps obvious but nevertheless important) that the universe acts as a giant magnifying glass for observing Quantum Geometry effects. Because of its expansion the cosmos, they point out, can serve as a classical detector of microscopic QG...

Hello. I am very curious about astronomy, and nothingness. I have a question though: Many people say the the universe is expanding, and say that beyond the universe, there is "nothing". What I think though, is that "something must exist beyond all that is already here, and this something I think...

I'm doing a proof, and near the last step I want to write the expression, \frac{d}{dt} \det{A(t)} = \lim_{\epsilon \to 0} \frac{\det{(A+\epsilon \frac{dA}{dt})} - \det{A}}{\epsilon} which produces the right answer, so I believe that it may be correct. This looks very much like a Taylor...

If gravity waves are supposed to propagate at c, how could the theoretical warp drive wave propagate at greater than c values? I have heard and red and understand that space itself can expand (and warp? ) faster than c and, in fact, that will happen in the future when space in between...

Homework Statement How many decimal expansions terminate in an infinite string of 9's? How many dont? The Attempt at a Solution If we have a number terminate with an infinite amount of 9's then it will be a rational number. So there would be countably many of these. And since...

So let's say a gas is doing some work against a piston and decreasing its internal energy. Does the gas also do work against gravity if it is expanding in the positive y direction? If so would it even be measurable? Thanks for the help

The book that I use (Concepts in Thermal Physics by S. and K. Blundell) states the second law in two ways. The way they state the Kelvin version is "no process is possible whose sole result is the complete conversion of heat into work." How does that fit in with the isothermal expansion of an...

I have no idea how to solve this problem. I feel like I need more information. For one the new height of the liquid should depends on the volume expansion of the glass tube. I am not given any dimensions of the glass tube other than it's height. Therefore if I am to believe that only the...

In Helmholtz original thesis On integrals of the hydrodynamical equations, which express vortex-motion, he mentioned in the first section that the change undergone by an arbitrary infinitesimal volume of water under the time dt is composed of three different motions. One of them is an expansion...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

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...

Homework Statement A thick walled insulating chamber contains ##n_1## moles of helium gas at a high pressure ##P_1## and temperature ##T_1##. It is allowed to leak out slowly to the atmosphere at a pressure ##P_o## through a small valve. Show that the final temperature of the ##n_2## moles...

what if the evolution of the universe over time and the expansion of space a series of binary events such that the same quantity of energy is divided geometrically from one into two, four, eight, sixteen pieces, such that each Planck time is equivalent to one such split and each Planck length is...

greetings gentlemen, i found this interpretation of red shift observations interesting: http://arxiv.org/abs/1303.6878/ my question is whether any new understanding can be gained by equating this increasing-mass explanation to the old expansion theory, if they both provide accurate...

1. Forgive me for this novice question, but can variation in the expansion rate of the universe, be partially explained by an increasing number of stars turning mass into energy, hence reducing gravitational attraction between galaxies, and at the same time applying radiation pressure (a small...

Is the expansion of the universe relative to our clocks and rulers? If so, it seems correct to include relativistic effects. And yet I am told that this is wrong. Regards, Jack Martinelli

hello In MTW excercise 22.6, given a fluid 4-velocity u, why the expression : ∇.u is called an expansion of the fluid world lines ? Is the following reasoning correct ? We know that the commutator : ∇BA - ∇AB is (see MTW box 9.2) is the failure of the quadrilateral formed by the vectors...

P(B/A) = P(A/B).P(B) / P(A) Later we expand P(A) as P(A/B).P(B) + P(A/B).P(B) ... B is complement of B I don't understand how we can expand P(A) like that. Doesn't that assume that A ℂ B?

I have an aluminum disc that is 15in in diameter and about 1in thick. I am going to put it in a 400°F oven and I need to know how much the diameter will expand. One of the engineers I work with(i'm an intern) is using what I found to be the linear expansion equation (ΔL/Li=αΔT)...so Length...

Hi, Could you help me to understand the following expansion I found in a book of qunatum mechanics. |1+a|^2=1+a+a*+... where a* is the complex conjugate of a

In my textbook W=∫p.dV is only proved for a syringe with a piston. This is quite easily done but the book never explains how it extrapolates to the general situation for a gas expanding in any deformable container. It seems the point is to prove dV= S.h where S is the surface area of a given...

I ask members here kindly for their assistance. I'm having some confusion over the process of integrating inequalities, in particular for obtaining the series expansion for the exponential function by integration. The text by Backhouse and Holdsworth (Pure Mathematics 2), shows the expansion of...

Hello :blushing: How to do expand this: (\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 - t)^2 where X(t_j)-X(t_{j-1}) = \Delta X_j to this: (\sum_{j=1}^{n}(\Delta X_j)^4 + 2*\sum_{i=1}^{n}\sum_{j<i}^{ }(\Delta X_i)^2(\Delta X_j)^2 -2*t*\sum_{j=1}^{n}(\Delta X_j)^2+t^2I get near the North Pole... but...

The Cauchy expansion says that \text{det} \begin{bmatrix} A & x \\[0.3em] y^T & a \end{bmatrix} = a \text{det}(A) - y^T \text{adj}(A) x , where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar. There is a proof in Matrix Analysis by Horn and...

"[F]ind the power-series expansion about the given point for each of the functions; find the largest disc in which the series is valid. 10. ##e^{z}## about ##z_{o} = \pi i##" (Complex Variables, 2nd edition; Stephen D. Fisher, pg. 133)$$f(z) = e^{z} = e^{z-a} \cdot e^{a} = e^{a} \cdot \sum...

Ahoy! I'm trying to approximate f'(r) for the following equation using matched asymptotic expansions -\frac{1}{2}\epsilon ff''=\left[\left(\epsilon+2r\right)f''\right]' where \epsilon \ll 1 and with the boundary conditions f(0)=f'(0)=0, \quad f'(\infty)=1 The inner expansion which...

I've long wondered about an assumption that we have today and I've never found a direct answer to my question. Presently we can observe that there is a direct proportionality between an object distance and the factor by which its light is redshifted. We deduce that this observation implies...

Homework Statement 1 mol of monoatomic ideal gas (temperature T1) is inside a cylinder with a moving piston (all are isolated). The initial external pressure on the piston is P1. at some point the external pressure is changed to (2/3)P1, the gas undergoes (irreversible) adiabatic expansion...

This is my 'virgin' attempt at intercourse in this venue, so please be roughly gentle. (I.E.; this is my first post, I'm a bit slow (intellectually) but willing to learn. I tend to be somewhat playful in how I use words. I hope you will forgive me if I step outside the boundaries of...

I have had a physics related question buzzing around my head for some time now, but have not been able to find the answer to it. I should state that have no training in physics, but have an interest in science and the universe. The question; if space is accelerating, what are the...

1. a steel plug has a diameter of 10.0 cm at 30 degrees celsius. at what temperature will the diameter be 9.986 cm? 2. a steel measuring tape is exactly 50,000 m long at 20 degrees celsius . (a) what is the length on hot summer when the temperature is 35 degrees celsius? (b) if such steel tape...

Hi everyone Homework Statement I want to find the multipole expansion of \Phi(\vec r)= \frac {1}{4\pi \epsilon_0} \int d^3 r' \frac {\rho(\vec r')}{|\vec r -\vec r'|} Homework Equations Taylor series The Attempt at a Solution My attempt at a solution was to use the Taylor series. I...

My school textbook says that discovery of the 1a supernova was what led to the understanding that the universe expansion is accelerating but doesn't hubble's equation already suggest that the universe would expand at an accelerating rate? V = Hd (V = velocity, H = hubble's constant, d =...

[a]Homework Statement 1.) A 2.00-liter aluminum cylinder at 5.00°C is filled to the brim with gasoline at the same temperature. If the aluminum and gasoline are warmed to 58.0°C, how much gasoline spills out? [Hint: Be sure to account for the expansion of the container. Also, ignore the...

Hi folks, If $e^x= \Sigma_{k=0}^\infty \frac{x^k}{k!}$ what do I evaluate $x$ at? How does the sigma notation tell me what to do with $x$? $$e^x= \Sigma_{k=0}^\infty \frac{x^k}{k!}\ = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} ... \text {ad infinitum}$$ Sorry, I just realized my error...

Homework Statement Suppose 100g of ethane (C2H6) expands isothermally at 350C from 50ml to 2L. Calculate the heat and work done by the system and the change in entropy if: i)The process is via a reversible path. ii)The process is non-reversible. I don't know how to answer ii) Homework...

I have some problems finding Taylor's expansion at infinity of f(x) = \frac{x}{1+e^{\frac{1}{x}}} I tried to find Taylor's expansion at 0 of : g(u) = \frac{1}{u} \cdot \frac{1}{1+e^u} \hspace{10 mm} \mbox{ where } \hspace{10 mm} u = 1/x in order to then use the known expansion of...

Good Morning Sirs, it seems to be surprisingly hard to get the numbers of a mystery: How fast expand hot rocket exhaust gases into empty space? Of course aside from its exit velocity! Does the expansion interfere with the impulse direction? I.e. is the impulse omnidirectional effective...