Taylor Definition and 878 Threads

Homework Statement For what values of x (or \theta or u as appropriate) do you expect the following Taylor Series to converge? DO NOT work out the series. \sqrt{x^{2}-x-2} about x = 1/3 sin(1-\theta^{2}) about \theta = 0 tanh (u) about u =1 Homework Equations The...

Homework Statement What degree Taylor Polynomial around a = 0(MacLaurin) is needed to approximate cos(0.25) to 5 decimals of accuracy? Homework Equations taylor series...to complicated to type out here remainder of nth degree taylor polynomial = |R(x)| <= M/(n+1)! * |x - a|^(n+1)...

Homework Statement f(x) = \frac{ln(3x)}{6x}, a = \frac{1}{3}, n=3 Find T3 Homework Equations Taylor Series - f(n)(x)/n! * (x-a)^n The Attempt at a Solution So, I isolated ln(3x) from 1/6x. I created the series based off of ln(3x). f(0)(x)=ln(3x) ->f(0)(1/3)=ln(3(1/3)) =0...

https://www.amazon.com/dp/189138922X/?tag=pfamazon01-20 Has anyone ever read this book? It looks like a bargain, good reviews, low price. What do you think of it? Is it a good mathematically oriented physics book?

Is a Fourier series essentially the analogue to a Taylor series except expressing a function as trigs functions rather than as polynomials? Like the Taylor series, is it ok only for analytic functions, i.e. the remainder term goes to zero as n->infinity?

Homework Statement Find the series solution for: y'=x^2-y^2,y(1)=1 Homework Equations The Attempt at a Solution I have correctly derived the series solution as: y(x)=1+(x-1)^2-\frac{(x-1)^3}{3}+\frac{(x-1)^4}{6}-... But I cannot get the book solution for the INTERVAL OF...

Homework Statement Find the Taylor Polynomial T2(x) (degree 2) for f(x) expanded about X0. f(x)=3x + cos(3x) X0= 0 Find the error formula and then find the actual (absolute) error using T2(0.6) to approx. f(0.6). The Attempt at a Solution As I've said on this forum before...

If a function can be represented by a Taylor series at x0, but only at this point, (radius of convergence = 0), is it considered analytic there?

I'm attempting to understand this notation (involving the Hessian) for the quadratic Taylor series for two variable. T_2 ( \tmmathbf{x}) = f ( \tmmathbf{a}) + \nabla f ( \tmmathbf{a}) \cdot ( \tmmathbf{x - a}) + \frac{1}{2} ( \tmmathbf{x - a}) \cdot H ( \tmmathbf{a}) \cdot (...

Can someone pls explain hot to compute a taylor expansion for f(x,y) using mathematica

Hey guys! I am attempting to do this problem and have been working with it for awhile now. Once again, it is an issue of the textbook not being very clear and making me more confused than ever. Sadly, our teacher is still MIA. Find the third Taylor polynomial P3(x) for the function f(x)=...

What's the Taylor expansion of F(x,y,z) in the neighborhood of (a,b,c)? Thank you

I can't work out how to calculate the Taylor series for \frac{1}{|R-r|} when R>>r, but they are both vectors. We were told to expand in r/R but I did the step below and I'm not sure where to go from there I got to \frac{1}{R \sqrt{1 - (2R.r)/R^2 + (r^2)/(R^2)}} I also know the result...

Using the taylor series result Vm / Vm - b = 1 + b / Vm + ... and the definition of hte compressibility factor Z = PVm / RT, derive an expression for the first virial coefficient in terms of a and b for the Berthelot equation of state.

Homework Statement Use Kepler's Third Law and a Taylor expansion to derive the following approximation for the orbital period of a satellite in low Earth orbit with a constant height h above the surface of the Earth. h << R_earth : P \approx P_{0}(1+3h/2R_{e}) Homework Equations Kepler's...

Expanding exp(hc / lambda*k_b * T) by Taylor series = 1 + hc /lambda*k_B * T +... But don't you take the derivative with respect to lambda? So I don't get how it would be this.

Is it correct to take the derivative of a taylor series the same as you would for a power series ie: sinx=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} \frac{d}{dx}(sinx)=cosx=\sum_{n=1}^{\infty}(-1)^n(2n+1)\frac{x^{2n}}{(2n+1)!} it seems as if it wouldn't be...

Is it correct that a taylor series does not exist for f(x)=tanh(x)/x and f(x)=ln(1+x)/x. I differentiated to f'''(x) and fn(0) and all equal zero.

Homework Statement Find the Taylor series about the point x = 0 for the function \frac{1}{3-2x^3} Homework Equations The Attempt at a Solution \frac{1}{3 - 2x^3} = \frac{1}{3(1 - \frac{2x^3}{3})} . Let u = \frac{2x^3}{3} . Then \frac{1}{3(1 - \frac{2x^3}{3})} = \frac{1}{3} \frac{1}{1 - u} =...

Homework Statement http://img243.imageshack.us/img243/4339/69855059.jpg I can't seem to get far. It makes use of the Exponentional Taylor Series: Homework Equations http://img31.imageshack.us/img31/6163/37267605.jpg The Attempt at a Solution taylor series expansions for cos...

Using P2(x,y), find a quadratic approximation to ln(1.25) to 4 decimal places. The original function is f(x,y)=ln(x2 + y2) and is about the point (1,0). I calculated P2 to be y2-x2+4x-3 however I don't know how to find a quadratic approximation. Do I just set say x=1 and y=.5? Any...

Homework Statement So I have the problem questiona dn my teachers solution posted below. I understand: f(xo) = sin pi/6 f '(xo) = cos pi/6 but i don't know how he gets them into fraction form with the SQRT of 3, it looks like some pythagoras but i don't really know how he did it...

ive got a question to ask I am working on taylor series and want to know f(x)=In(3+x) and g(x)=In (1+x) by writing In(3+x)=In3+In(1+1/3x) im asked to use substitution in one off the standard taylor series given in the course.to find about 0 for f explicitly all...

First of all if i have a function with all negative terms is it possible to determine its convergence simply by factoring the negative one, treating the other terms as a positive series determine its convergence then assume that multiplying by the constant negative one will not change its...

Homework Statement (Goldstein 3.3) If the difference \psi - \omega t in represented by \rho, Kepler's equation can be written: \rho = e Sin(\omega t + \rho) Successive approximations to \rho can be obtained by expanding Sin(\rho) in a Taylor series in \rho, and then replacing \rho...

Find the function that has the following Taylor series representation: \sum^{\infty}_________{m=0}\frac{(m+s)^{-1}x^{m}}{m!} Where s is a constant such that 0<Re(s)<1. Any ideas?

Homework Statement 1. Use Taylor's Theorem to determine the accuracy of the approximation. arcsin(0.4) = 0.4 + \frac{(0.4)^{3}}{2*3}} 2. Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value fo x to be less...

I'm reading a paper on tissue cell rheology ("Viscoelasticity of the human red blood cell") that models the creep compliance of the cell (in the s-domain) as J(s) = \frac{1}{As+Bs^{a+1}} where 0\leq a\leq 1. Since there's no closed-form inverse Laplace transform for this expression, they...

When approximating a function with a Taylor series, I understand a series is centered around a given point a, and converges within a certain radius R. Say for a series with center a the interval of convergence is [a-R, a+R]. Does this imply that: 1. There also exists a Taylor series expansion...

Hello, I am looking for a resource (preferably a textbook) to help me with nonlinear, multivariable functions and working through taylor series expansions of them. My calculus book only covers single variable expansions unfortunately. Thanks

Question about Taylor series and "big Oh" notation Can someone please explain WHY it's true that e^x = 1 + x + \frac{x^2}{2} + \mathcal{O}(x^3) I'm somewhat familiar with "big Oh" notation and what it stands for, but I'm not quite sure why the above statement is true (or statements...

Homework Statement (a) Give Taylor Polynomal of order 4 for ln(1+x) about 0. (b) Write down Tn(x) of order n by looking at patterns in derivatives in part (a), where n is a positive integer. (c) Write down the remainder term for the poly. in (b) (d) How large must n be to ensure Tn gives a...

Homework Statement f(x) = \frac{1-cos(X^2)}{x^3} which identity shoud i use? and tips on this type of questions? once i can separate them, then i'll be good thanks!

you know this, right? f(x) = \sum^{\infty}_{k=0} \frac{f^{(k)}(x_0) (x-x_0)^k}{k!} for an analytic function, at x0 = 0, you have to say that 0^0 equals 1 for the constant term. if 0^0 is indeterminate then how can you just say it's 1 in this case?

Homework Statement I want to know that how to calculate the required number of terms to obtain a given decimal accuracy in two variable Taylor series . In one variable case i know there is an error term R(n)=[ f(e)^(n+1)* (x-c)^(n+1)] / (n+1)! where 'e' is...

[b]1. Hi, I am new to taylor series expansions and just wondered if somebody could demonstrate how to do the following. Find the Taylor series of the following functions by using the standard Taylor series also find the Radius of convergence in each case. 1.log(x) about x=2...

Homework Statement (a) Use Taylor's theorem with the Lagrange remainder to show that log(1+x) = \sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}x^{k} for 0<x<1. (b) Now apply Taylor's theorem to log(1-x) to show that the above result holds for -1<x<0. Homework Equations Taylor's...

In this: http://www.math.tamu.edu/~fulling/coalweb/sinsubst.pdf It says that to find the Taylor series of sin(2x + 1) around the point x = 0, we cannot just substitute 2x+1 into the Maclaurin series for sinx because 2x + 1 doesn't approach a limit of 0 as x approaches 0. It says we have...

1. The problem \statement, all variables and given/known data Estimate the error involved in using the first n terms for the function F(x) = \int_0^x e^{-t^2} dt Homework Equations The Attempt at a Solution I am using the Lagrange form of the remainder. I need to know the n+1 derivative of...

[b]1. Use Taylor's expansion about zero to find approximations as follows. You need not compute explicitly the finite sums. (a) sin(1) to within 10^-12; (b) e to within 10^-18: [b]3. I know that the taylor expansion for e is e=\sum_{n=1}^{\infty}\frac{1}x^{n}/n! and I aslo know that...

Homework Statement A water wave has length L moves with velocity V across body of water with depth d, then v^2=gL/2pi•tanh(2pi•d/L) A) if water is deep, show that v^2~(gL/2pi)^1/2 B) if shallow use maclairin series for tanh to show v~(gd)^1/2 Homework Equations Up above [b]3. The...

Find P5(x), the 5th order Taylor series, of sin (x) about x = 0. Hence find the 4th order Taylor series for x sin (2x) about x = 0. In this question why is it required to find the 5th order taylor series of sin(x) to find the 4th order taylor series of xsin(2x)?

Sorry, the title should be: geometric intepretation of moments My question is: does the formula of the moments have a geometrical interpreation? It is defined as: m(p) = \int{x^{p}f(x)dx} If you can't see the formula it is here too: http://en.wikipedia.org/wiki/Moment_(mathematics) with c=0...

Homework Statement Could someone please explain how the taylor expansion of 1/(r-r') turns into ( 1/r+(r'.r)/r^3 + (3(r.r')^2-r^2r'^2)/2r^5 +...) Homework Equations The Attempt at a Solution

How do you Taylor expand e^{i \vec{k} \cdot \vec{r}} the general formula is \phi(\vec{r}+\vec{a})=\sum_{n=0}^{\infty} \frac{1}{n!} (\vec{a} \cdot \nabla)^n \phi(\vec{a}) but \vec{k} \cdot \vec{r} isn't of the form \vec{r}+\vec{a} is it?

Homework Statement I'm having a hard time following a taylor expansion that contains vectors... http://img9.imageshack.us/img9/9656/blahz.png http://g.imageshack.us/img9/blahz.png/1/ Homework Equations The Attempt at a Solution Here's how I would expand it: -GMR/R^3 -...

Homework Statement Prove if t > 1 then log(t) - \int^{t+1}_{t}log(x) dx differs from -\frac{t}{2} by less than \frac{t^2}{6} Homework Equations Hint: Work out the integral using Taylor series for log(1+x) at the point 0 The Attempt at a Solution Using substitution I get...

Homework Statement Let f be a function with derivatives of all orders and for which f(2)=7. When n is odd, the nth derivative of f at x=2 is 0. When n is even and n=>2, the nth derivative of f at x=2 is given by f(n) (2)= (n-1)!/3n a. Write the sixth-degree Taylor polynomial for f about...

Homework Statement The Taylor polynomial of degree 100 for the function f about x=3 is given by p(x)= (x-3)^2 - (x-3)^4/2! +... + (-1)^n+1 [(x-3)^n2]/n! +... - (x-3)^100/50! What is the value of f^30 (3)? D) 1/15! or E)30!/15! Homework Equations The Attempt at a...

an idea i had: factorizing taylor polynomials Can any taylor polynomial be factorized into an infinite product representation? I think so. I was able to do this(kinda) with sin(x), i did it this way. because sin(0)=0, there must be an x in the factorization. because every x of...