Taylor expansion Definition and 166 Threads

can anybody help me with this integration? Integral of e to the -2x times x squared dx it expands to 1/4, but I'm not sure how to start.

\biggl(-\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} +\mathcal{O}(x^8)\biggr)-\frac12\biggl(-\frac{x^2}2+\frac{x^4}{24}+\mathcal{O}(x^6)\biggr)^2+\frac13\biggl(-\frac{x^2}2+\mathcal{O}(x^4)\biggr)^3 + \mathcal{O}(x^8)\\ & =-\frac{x^2}2 + \frac{x^4}{24}-\frac{x^6}{720} - \frac{x^4}8 +...

Homework Statement I need to find the bloch vector for the density matrix \frac{1}{N}\exp{-\frac{H}{-k_bT}} where the Hamiltonian is given by H=\hbar\omega\sigma_z. The Attempt at a Solution I can break the Taylor series of exp into odd and even terms because sigma z squared is the...

Homework Statement For g=Hf = sin (f), use a Taylor expansion to determine the range of input for which the operator is approximately linear within 10 % Homework Equations The taylor series from 0 to 1 , the linearization, is the most appropriate equation The Attempt at a Solution...

Homework Statement Linearize the system operator illustrated below by applying a Taylor series expansion. f(t) ----> e^f(t) -----> g(t) Homework Equations I only find the general form of a taylor series relevant. g(x)= sum (0,infinity) of [f^n*(a)*(x-a)^n]/n! The system is...

This is just part of a larger problem, but I have a basic equation r'=k-g*r, where k and a start out as constants, but then I need to treat everything as if it can vary slightly from the average. For this, I set r=r_ave+dr, g=g_ave+dg, and k=k_ave+dk. Now I need to work these into the first...

Hi I wonder if there is a simpler way to obtain the first three non-zero terms of Taylor Expansion for the function \frac{Ln(1+x)}{1-x} about x=0? I differentiated it directly, but it was such a nightmare to do:mad: . So I am wondering if there is a simpler way to do it?

Homework Statement I have the following question to answer: Show that (X^2/h^2)*((1/2*y1) - y2 + (1/2*y3)) + (X/h)*((-1/2 y1)+(1/2 y3))+y2 (sorry about the format) is equal to (taylor expansion): y = y2+(x(dy/dx)¦0 + (x^2/2*((d^2)y)/(dx^2))¦0 Homework Equations also given in...

Hey Everyone. I'm ALMOST finished this problem... To spare you the long story, I need to take the difference between an gravitational acceleration, and the same gravitational acceleration at a slightly larger height. The two functions are a(r) and a(r+d), where d is very small Now... VERY...

I'm having trouble determining the order of the pole of [exp(iz) - 1]/((z^2) + 4) at z=2i I know I can't just expand the exponential as 1 + iz + [(iz)^2]/2 ... because this formula only works near the origin. Can I still use Taylor's theorem to find the expansion at z=2i (i.e does...

the problem reads develop expansion of ln(1+z) of course I just tried throwing it into the formula for taylor expansions, however I do not know what F(a) is, the problem doesn't specify, so how can I use a taylor series?

Can anyone please give me a hint on any of the following Taylor expansions? That would be so helpful! for all three: Find the first non-zero term in the Taylor series about x = 0 problem 1 \frac{1} {sin^2x} - \frac{1} {x^2} everytime I differentiate the result is zero...so that...

I am supposed to find an approximation of this integral evaluated between the limits 0 and 1 using a taylor expansion for cos x: \int \frac{1 - cos x}{x}dx and given cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}... i should get a simple series similar to this for...

Hi How do you expand (1-exp(-1))^-1 as Taylor series Callisto

V = 2\pi \sigma(\sqrt{R^2+a^2}-R) Show that for large R, V \approx \frac{\pi a^2 \sigma}{R} I figured if I could develop the MacLaurin serie with respect to an expression in R such that when R is very large, this expression is near zero, then the first 1 or 2 terms should be a fairly...

basic taylor expansion... Hi, could some one explain how i could use the taylor series to expand out: f(x)= 1/sqrt(1-x^2) Any help would be appreciated, thanks.