Taylor Definition and 849 Threads

Homework Statement z is a complex number. find the taylor series expansion for g(z)=1/(z^3) about z0= 2.in what domain does the taylor series of g converge. z0 is z subscript 0 Homework Equations The Attempt at a Solution I wrote g(z)=1/(z^3) = 1/(2+(z^3)-2) = (1/2)*1/(1+(z^3...

Ok, we are asked to determined the degree of the the taylor polynomial about c =1 that should be used to approximate ln (1.2) so the error is less than .001 the book goes throught the steps and arrives at: |Rn(1.2)| = (.02)^(n+1)/(z^(n+1)*(n+1) but then, it states that...

[SOLVED] Taylor Series Question I have to find the Taylor series of \frac{3}{z-4i} about -5. Therefore, we want the series in powers of z+5. Now, following the textbook it appears that we want to get this in a form that resembles a geometric series so that we can easily express the Taylor...

Homework Statement http://img99.imageshack.us/img99/9044/tayloriq0.th.jpg Homework Equations ? The Attempt at a Solution I have no idea - please help...

Homework Statement Determine the Taylor Series for f(x)=sinx about the center point c=pi/6Homework Equations pn(x) = f(c) + f'(c)(x-c) + f''(c)(x-c)^2/2! + f'''(c)(x-c)^3/3! + ...The Attempt at a Solution f(pi/6) = 1/2 f'(pi/6) = \sqrt{3}/2 f''(pi/6) = -1/2 f'''(pi/6) = -\sqrt{3}/2 f(4)(pi/6)...

I'm unclear on what they are asking in this homework problem. Suppose we know a function f(z) is analytic in the finite z plane apart from singularities at z = i and z=-1. Moreover, let f(z) be given by the Taylor series: f(z)=\displaystyle\sum_{j=0}^{\infty}a_{j}z^{j} where aj is...

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 Use the "Three Term" Taylor's approximation to find approximate values y_1 through y_20 with h=.1 for this Initial Value Problem: y'= cosh(4x^2-2y^2) y(0)=14 And write a computer program to do the grunt work approximation Homework Equations The Attempt...

How does one prove taylor series? Is it proven the same way as Maclaurin's Series(Which i know is a special case of taylor series) f(x)=A_0+A_1x+A_2x^2+A_3x^3+... f(\alpha)=A_0+A_1\alpha+A_2(\alpha)^2+A_3(\alpha)^3+... this kinda doesn't seem like a good way to prove it...as that is how I...

Is there any nice trick for finding the Taylor polynomial of a composition of 2 functions, both of which can be expressed as taylor polynomials themselves? For example, finding the taylor polynomial for e^{\cos x}. Thanks.

Homework Statement I have E(v) = (m*c^2)/sqrt(1-v^2/c^2). I also have a second-order Taylor-polynomial around v = 0, T_2_E, which is mc^2+½mv^2. I have to use Taylors formula with restterm to show that E is bigger than T_2_E for all v in the interval [0,c). The Attempt at a Solution...

I have E(v) = (mc^2)/(sqrt(1-(v^2/c^2)). I have found the second-order Taylor-polynomial for v=0, and I get: T_2_E(v) = mc^2 + ½mv^2. My teacher asks me, why this equation must be true - what is so special about the second order Taylor-polynomial for v = 0 for E(v)?

Homework Statement find an interval I such that the tangent line error bound is always less than or equal to 0.01 on I f(x) = ln(x) b = 1 The Attempt at a Solution so basically, i found the tangent line approximation at b = 1, which is t(x) = x -1. From there though, i have no idea...

Homework Statement Consider f(x) = 1 + x + 2x^2+3x^3. Using Taylor series approxomation, approximate f(x) arround x=x0 and x=0 by a linear function Homework Equations The Attempt at a Solution This is the first time that I have seen Taylor series and I am totally lost on how to...

Homework Statement I've been asked to: Use the real Taylor series formulae e^{x} = 1 + x + O(x^{2}) cos x = 1 + O(x^{2}) sin x = x(1 + O(x^{2})) where O(x^{2}) means we are omitting terms proportional to power x^{2} (i.e., \lim_{x\rightarrow0} \frac{O(x^{2})}{x^{2}} = C where C is a...

I was going through the derivation of the Taylors series in my book (Engineering Mathematics by Jaggi & Mathur), and there was one step that escaped me. They proved that the derivative of f(x+h) is the same wrt h and wrt (x+h). If someone could explain that, Id be really grateful.

Hi I have some questions. If you're doing a MacLaurin expansion on a function say sinx or whatever, if you take an infinite number of terms in your series will it be 100% accurate? So will the MacLaurin series then be perfectly equal to the thing you're expanding? Also I don't really...

Homework Statement Write down the Taylor Polynomial of degree n of the function f(x) at x=0 Homework Equations f(x) = ln(1-x) The Attempt at a Solution f(x) = ln(1-x) f'(x) = (-1)((1-x)^(-1)) f``(x) = (-1)((1-x)^(-2)) f```(x) = (-2)((1-x)^(-3)) f````(x) =...

Homework Statement ln(1+x)=x-\frac{1}{2}x^2+ \frac{1}{3}x^3-\frac{1}{4}x^4+\frac{1}{5}x^5-... -1<\ x\ <1 Is there a Taylor polynomial for ln(1-x) for -1< x <1, if so how would I go about working it out from the above? This is not really a homework question just a thought I had, as they do it...

Homework Statement Can someone explain big O notation to me in the context of taylor series? For instance, how do you know that sint t = t - t^3/(3t)! + O(t^5) as t -> 0? Does that hold when t -> infinity as well? Is there a generalization of this rule? Is it derived from the...

I've stumbled upon what might be a geometrical interpretation of Taylor's series for sine and cosine. Instead of deriving the Taylor's series by summing infinite derivatives over factorials, I can derive the same approximation from purely geometrical constructs. I'm wondering if something...

Homework Statement Hi everyone, determine a Taylor Series about x=-1 for the integral of: [sin(x+1)]/(x^2+2x+1).dx Homework Equations As far as I know the only relevant equation is the Taylor Series expansion formula. I've just started to tackle Taylor Series questions and I've been...

Homework Statement Expand cos z into a Taylor series about the point z_0 = (pi)/2 With the aid of the identity cos(z) = -sin(z - pi/2) Homework Equations Taylor series expansion for sin sinu = \sum^{infty}_{n=0} (-1)^n * \frac{u^{2n+1}}{(2n+1)!} and the identity as given...

I was just curious why when doing a taylor series like xe^(-x^3) we must first find the series of e^x then basically work it from there, why can't we instead do it directly by taking the derivatives of xe^(-x^3). But doing it that way doesn't give a working taylor series why is this so?

My exam is coming up, I have 2 questions on infinite series. Any help is appreciated!:smile: Quesetion 1) http://www.geocities.com/asdfasdf23135/calexam1.JPG For part a, I got: g(x)= Sigma (n=0, infinity) [(-1)^n * x^(2n)] For part b, I got: x ∫ tan^-1...

Homework Statement Approximate f by a Taylor polynomial with degree n at the number a. f(x) = x^(1/2) a=4 n=2 4<x<4.2 (This information may not be needed for this, there are two parts but I only need help on the first) Homework Equations Summation f^(i) (a) * (x-a)^i / i! The Attempt at...

In the Acknowledgements of "Exploring Black Holes" by Taylor and Wheeler they mention that a solutions manual was created by G.P. Sastry and several students. Does anyone know whether it is possible to get a hold of this solutions manual for us self learners. It would be a great help...

Taylor formula. Need help! we have f(a+h), where a-is a point, and h is a very small term, h->0. And we have the formula to evaluate the function y=f(x), around the point a, which is f(a+h)=f(a)+f'(a)h+o(h) --------(*) however when we want to take in consideration o(h) this formula does...

I don't get how these two forms of the taylor series are equivalent: f(x+h)= \sum_{k=0}^{\infty} \frac{f^k(x)}{k!} h^k f(x) = \sum_{k=0}^{\infty} \frac{f^k(0)}{k!}x^k The second one makes sense but I just can't derive the first form using the second. I know its something very simple...

Homework Statement Give the Taylor Series for exp(x^3) around x = 2. Homework Equations f(x) = Sum[f(nth derivative)(x-2)^n]/n! The Attempt at a Solution I know the solution for e^x but can't seem to find a formula for the nth derivative of exp(x^3) around x = 2. Thanks for...

Let ƒ be the function given by f (x) = e ^ (x / 2) (a) Write the first four nonzero terms and the general term for the Taylor series expansion of ƒ(x) about x = 0. (b) Use the result from part (a) to write the first three nonzero terms and the general term of the series expansion about x = 0...

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...

1) Let f(x) = (x^3) [cos(x^2)]. a) Find P_(4n+3) (x) (the 4n + 3-rd Taylor polynomial of f(x) ) b) Find f^(n) (0) for all natural numbers n. (the n-th derivative of f evaluated at 0) I know the definition of Taylor polynomial but I am still unable to do this quesiton. I tried to find the...

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...

Homework Statement I'm trying to make the nth degree taylor polynomial for f(x)=sqrtx centered at 4 and then approximate sqrt(4.1) using the 5th degree polynomial I know that the polynomials are found using the form: P(x)= f(x)+f'(x)x+f''(x)x^2/2factorial...f^n(x)x^n/nfactorial so...

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...

Ok there's something I don't get. I know for instance that the linear polynomial for say f = 91 + 2x + 3y + 8z + Quadratic(x, y, z) + Cubic(x, y, z) ... is 91 + 2x + 3y + 8z if the base point is (0, 0, 0). This is pretty clear. What I don't get is why when you take the base point to be say (1...

i'm having a hard time understanding taylor series and why it works and how it works. if someone could please explain it to me that would be great. My teacher explained it in class but he goes so fast that i have no idea what he's saying. he did give us some practice problems but if i have no...

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?

hi everyone, I am just learning the taylor series at school. I am slightly confused. in my textbook, one of hte exercises is to find hte nth degree taylor polynomial of x^4 about a=-1. n is 4 in this case so this gives me a long polynomial. i understand that inputting any x value into this...

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...

any insight to this question? .. i mean.. usually people just do up to order 2.. find the taylor polynomial of order 3 based at (x, y) = (0, 0) for the function f(x, y) = (e^(x-2y)) / (1 + x^2 - y) how large do you have to take k so that the kth order taylor polynomial f about (0, 0)...

The electric potential V at a distance R along the axis perpendicular to the center of a charged disc with radius a and constant charge density d is give by V = 2pi*d*(SQRT(R^2 +a^2) - R) Show that for large R V = pi*a^2*d / R This is what I have done so far... V = 2pi*d *...

I am supposed to prove using taylor series the following: \frac{d^2\Psi}{dx^2} \approx \frac{1}{h^2}[\Psi (x+h) - 2\Psi(x) + \Psi (x-h)] where x is the point where the derivative is evaluated and h is a small quantity. what i have done is used: f(x+h)= f(x) + f'(x) h +...

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...

Let be an analytic function f(x,y) so we want to take its Taylor series, my question is if we can do this: -First we expand f(x,y) on powers of y considering x a constant so: f(x,y)= \sum_{n=0}^{\infty}a_{n} (x)y^{n} and then we expand a(n,x) for every n into powers of x so we have...

I have got a question here that puzzles me. How do I use TAYLOR SERIES to find the 2005th derivative for the function when x=0 for the following function: f(x) = inverse tan [(1+x)/(1-x)] Part (1) I was hinted that differentiating inverse tan x is = 1/(1+x^2). Part (2) After which, I need to...

In textbooks these polynomials are not normally presented as an infinite series (the single variables are). What is the reason for this and are they equally allowed to be in infinite series form hence infinite order just like the single variable Taylor Polynomials? Or are there more issues about...

Hi Guys, I have an assigment which I would very much appreciate if You would tell if I have done it correct :) Use the Taylor Polynomial for f(x) = \sqrt(x) of degree 2 in x = 100. To the the approximation for the value \sqrt(99) First I find the Taylor polynomial of degree 2...

With a simple ODE like \frac{ds}{dt} = 10 - 9.8t and you're given an initial condition of s(0) = 1, when doing the approximation would s'(0) = 10 - 9.8(0), s'' = ... etc?