Taylor Definition and 849 Threads

Hello folks, I have this function, un complex numbers \frac{1}{(1+z^2)} I know that the Taylor serie of that function is \frac{1}{(1+z^2)} = \sum (-1)^k.z^(2.k)

Homework Statement I'm given that the function f(x) is n times differentiable over an interval I and that there exists a polynomial Q(x) of degree less than or equal to n s.t. \left|f(x) - Q(x)\right| \leq K\left|x - a\right|^{n+1} for a constant K and for a \in I I am to show that Q(x)...

Does anyone know a proof of taylor formula (actually I am looking for proof for maclaurin series but guess it is the same) without using derivation rules for polynomials?

I am reading through a worked example of the Taylor series expansion of Sinh(z) about z=j*Pi The example states: sinh(j*Pi)=cos(Pi)*Sinh(0) +jcosh(x)sin(y) I am unsure of this relation. I understand why the x terms are zero but don't know the relation to expand sinh. Can anyone shed...

For #4, I'm mostly confident I did it correctly. In determining the error, we're supposed to find the maximum absolute value on an interval I. I set I = (0,2pi). Is that right? http://i111.photobucket.com/albums/n149/camarolt4z28/4-1.png For #5...

Homework Statement Find the Taylor series expansion for f(x)=x*e^(-x^2) about x = -1 Homework Equations The Attempt at a Solution I have tried replacing x with (x-1) and f(x-1) = (x-1)*e^(-(x-1)^2). Consider the power series for e^(-(x-1)^2) about x = 0, f(x-1) =...

Find the Taylor series expansions for f(x)=x*e^(-x^2) about x = -1 -(1/E) - (x + 1)/E + (x + 1)^2/E + (5 (x + 1)^3)/(3 E) + (x + 1)^4/( 6 E) - (23 (x + 1)^5)/(30 E) - (29 (x + 1)^6)/(90 E) + ( 103 (x + 1)^7)/(630 E)... This is the answer from Mathematica but i don't know how it goes. Can...

I'm looking at the series published @ Wikipedia: http://en.wikipedia.org/wiki/Inverse_hyperbolic_function There is a series for arsinh, which I was able to derive with no problem - basically take the derivative of arsinh, which is a radical, then apply the general binomial expansion, which...

Homework Statement Find the sums of the following series: S1=1+(x^3)/(3!)+(x^6)/(6!)+... S2=x+(x^4)/(4!)+(x^7)/(7!)+... S3=(x^2)/(2!)+(x^5)/(5!)+(x^8)/(8!)+... Homework Equations Perhaps Taylor series? The Attempt at a Solution I spotted that adding S1+S2+S3=e^x, but I don't...

Homework Statement What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places? Homework Equations R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial) The Attempt at a Solution I have attached my attempt. I am stuck on the...

Homework Statement Find the Taylor series expansions for f(z) = −1/z^2 about z = i + 1. Homework Equations The Attempt at a Solution I'm just not sure what format I'm supposed to leave it in. Is it meant too look like this: f(z)=f(i+1)+f'(i+1)(x-i-1)... or this Ʃ\frac{1}{n!}f^{(n)}(1+i) *...

Hello! I am wondering if someone could let me know if my understanding is right or wrong. The Taylor series gives the function in the form of a sum of an infinite series. From this an approximation of the change in the function can be derived: f_{a} and f_{a,a} are the first and second...

The equation starts at B and this is my attempt. As you can see it soon complicates and doesn't look like what t should since I already know what the Taylor series of his function should look like. Is there some clever trick to it that I am missing? PS the series is centred around x = 0...

Show that, with an appropriate choice of constant c, the taylor series of (1+cx)ln(1+x) has terms which decay as 1/n^2 I know that ln(1+x) decays as 1/n, but I don't know how to show the above. Please help. Thanks in advance

Homework Statement The following is a modification of Newton's method: xn+1 = xn - f(xn) / g(xn) where g(xn) = (f(xn + f(xn)) - f(xn)) / f(xn) Homework Equations We are supposed to use the following method: let En = xn + p where p = root → xn = p + En Moreover, f(xn) = f(p + En) = f(p) +...

Currently, I'm doing some self studying on series, and I'm a bit confused regarding c (the value that the series is expanded about). For example, does the Maclaurin series expansion of Sin(x) and the Taylor series of Sin(x) about c = 1 both converge to Sin(x)? If so, what does the value...

I have to prove that \cos(x) = 1 - \frac{x^2}{2} + O(x^4) (x \to 0) My ugly attempt: \lim_{x \to 0} \frac{\cos(x) - 1 + \frac{x^2}{2}}{x^4} \lim_{x \to 0} \frac{\cos(x) - 1}{x^4} + \frac{1}{2x^2} \lim_{x \to 0} \frac{\sin(x)}{4x^3} + \frac{1}{2x^2} \lim_{x \to 0}...

Hello, I'm new here, nice to meet you guys i was in class today and just didn't understand the taylor polynomial approximation, the professor started out approximating a function by polynomials of degree N, he first showed us how a linear polynomial was a crude approximation of the function...

Ive attached the problem and my work in the pic. Questions: Am I even applying the taylor polynomial the correct way? (I never learned taylor series, but I was supposed to be taught in the pre-requisite class) Am I suppose to plug in c=4? I am not so sure about how the U4(t) works...

what does it mean to say taylor expansion of ex centered at 0? does it mean that the sum of the expansion will give me the value that the function ex will take when x = 0 ? so its e0 = 1? also, how do we know what value to center on? because as i encounter taylor series in my...

Hello all, I understand that the taylor expansion for a multidimensional function can be written as f(\overline{X} + \overline{P}) = f(\overline{X}) + \nabla f(\overline{X}+t\overline{P})(\overline{P}) where t is on (0,1). Although I haven't seen that form before, it makes sense...

Homework Statement Hello, I'm in the middle of solving for the Taylor series of the function: f(x)=sin(2x)ln(1-x) up to n = 4. The Attempt at a Solution So far, I've been strictly taking its derivatives until I reach the fourth. It's becoming a very long process considering it's...

Homework Statement Prove that 0.493948<\int_0^{1/2} \frac{1}{1+x^4} dx<0.493958Homework Equations This chapter is about Taylor Polynomials, and specifically this section deals with Taylor's formula with remainder: f(x)=\sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k + E_n(x) The general formula for...

Homework Statement For f(z) = 1/(1+z^2) a) find the taylor series centred at the origin and the radius of convergence. b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius. Homework...

upper bound of taylor! f(x) is two times diff. function on (0, \infty) . \lim\limits_{x\rightarrow \infty}f(x) = 0 satisfy. M=\sup\limits_{x>0}\vert f^{\prime \prime} (x) \vert satisfy . for each integer L , g(L) = \sup\limits_{x\geq L} \vert f(x) \vert, and h(L) = \sup\limits_{x\geq L} \vert...

Given a partial differential equation, how would one go about implementing the forward finite difference method to the Taylor series?

Homework Statement https://skydrive.live.com/?cid=6b041751c72e14ad#!/?cid=6b041751c72e14ad&sc=photos&uc=3&id=6B041751C72E14AD%21149!cid=6B041751C72E14AD&id=6B041751C72E14AD%21154&sc=photos The Attempt at a Solution...

Homework Statement I posted this already but decided to revive this thread since I re-worked the problem. Consider dy/dx=x+y, a function of both x and y subject to initial condition, y(x0)=y0. Use Taylor series to determine y(x0+\Deltax) to 4th order accuracy. Initial condition: x0=0...

Homework Statement This shouldn't be so hard to do I guess, but I just cannot figure it out. The problem statement: Prove that the special form of the discrete Laplacian operator in radial coordinates acting on a grid function u_{l,m} at the central grid point l=0, m=0, given by...

Hi By some googling it seems like there exist some kind of expansion of the Taylor series for statistical functionals. I can however, not sort out how it is working and what the derivative-equivalent of the functional actually is. My situation is that I have a functional, say \theta which...

Homework Statement Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x) Homework Equations The Attempt at a Solution Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x) f(x) =...

New Question (Changed Old one) - Taylor Polynomial - Upper Bound for Absolute Error Homework Statement (a) Find the 3-rd degree Taylor polynomial of sin(pix) centered at x=1. (b) Use (a) to approximate sin(1.1*pi) (c) Use the remainder term to find an upper bound for the absolute error in...

Homework Statement Hi, I'm really struggling with trying to come up with the error bound when doing taylor series problems Use the reaminder term to estimate the absolute error in approximating the following quantitites with the nth-order Taylor Polynomial cnetered at 0. Estimates are...

Homework Statement f(x,y) = ln(3y-8x) Derive the first and second order Taylor polynomial approx, L(x,t) and Q(x,t), for T(x,T) about the point (1,1) Homework Equations -None- The Attempt at a Solution I do not understand what the question wants, nor do i want a solution. I...

hello, I'm examinating the theorem of power series, specially taylor series I know a function f(x) can be written as a series of polynomials. but using the taylor series it says that the convergence of that function is about a point a by using the Maclaurinseries a = 0 , so examinating...

So I'm computing a second order Taylor series expansion on a function that has multiple variables. So far I have this I(x,y,t)=dI/dx(change in x)+dI/dy(change in y)+dI/dt(change in t)+2nd order terms Would it still be a better approximation than just he first order if I included some...

I'm trying to learn about finite difference methods to solve differential equations. I'm using Advanced Engineering Mathematics 9th Ed., and in explaining Euler's method he claims the following Taylor series: y(x+h) = y(x) + hy'(x) + \dfrac{h^2}{2}y''(x) + \cdots He then truncates that...

Now although this is a problem for my EE course, it is more of a calculus question so I figured I would receive the best answers by posting it in this section. I have just started on the problem but could use some input on my thoughts. So here we go (there are two parts): (problem screenshot is...

When a Taylor Series is generated from a functions n derivatives at a single point, then is that series for any value of x equal to the original function for any value x ? For example graph the original function (x) from x= 0 to x = 10. Now plug into the Taylor Expansion for x , values...

For retarded scalar potential of arbigtrary source around origin: V(\vec r, t) = \frac 1 {4\pi\epsilon_0}\int \frac { \rho(\vec r\;',t-\frac {\eta}{c}) }{\eta} d\;\tau' \;\hbox { where }\;\eta =\sqrt{r^2 + r'^2 - 2 \vec r \cdot \vec r\;' } Where \;\vec r \; point to the field point where V...

Homework Statement series expansion at c=2 of ln(x^2+x-6) Homework Equations The Attempt at a Solution After substituting y= x -2 we get ln(y^2+5y) = ln(y) + ln(y+5) but I am not kinda sure how to use the taylor series of ln(1+x)...

I have a question about Taylor expanding functions. For both cases I can't get my head around why things are the way they are. I just don't see how one would perform Taylor expansions like that. The first: The starting point of a symmetry operations is the following expansion: f(r+a) = f(r)...

I read wikipedia article also but I can't find the proof of taylor series and from where it came from??

Dear all, This question is close to the post "Laplace transform of a Taylor series expansion" in PhysicsForums.com, dated Jul06-09. This is my problem: Consider the Laplace transform F(s) = 1 / ( s - K(s) ) , where K(s) = -1/2 + i/(2*Pi) * ln[ ( Lambda - (b+i*s) )/( b + i*s...

On p35 of Jackson's Classical Electrodynamics 3rd Edition, the author gives the expansion of the charge density \rho(\mathbf{x'}) around \mathbf{x'}=\mathbf{x} as \rho(\mathbf{x'}) = \rho(\mathbf{x}) + \frac{r^2}{6}\nabla^2\rho + ... where r = |\mathbf{x} - \mathbf{x'}| My question is...

Usually to do the remainder we take Rn(x) = (f differentiated n+1 times at a ).(x-c)n+1/(n+1)!, but when my function is sin(x) do i take (f differentiated 2n+2 times at a ).(x-c)2n+2/(2n+2)!? Thanks

Homework Statement How many terms of the taylor series of the cosine function about c = 0 are needed to calculate cosine 2 to an accuracy of 1 / 10000 The Attempt at a Solution I have said that |Rn(2)| = |cosn+1(a) 2n+1/(n+1)!|<2n+1/(n+1!) Now i can't do it ...

I have this translation operator T(a) that acts on a function y(x) and causes the transformation T(a)y(x) = y(x+a). I am supposed to be "expanding y(x+a) as a taylor series in a" to show that T(a)=eipa, where p is the operator p = -i.d/dx] So, I've started out with the general equation for the...

What does it mean to have a taylor expansion of a gradient (vector) about the position x? I.e. taylor expansion of g(x + d) where g is the gradient and d is the small neighborhood.

I consider an array of lattice points and construct a vector at each lattice points. How to convert this discrete system into a continuum one by using the Taylor series expansion by considering the lattice distance say \lambda? thanks in well advance?