Taylor series Definition and 480 Threads

Hello, Is there any place I can find the equation for the Taylor expansion of a functional around a function ?? Particularly, I want something like: f[x(t)] = f[\hat{x}(t)] + (f[\hat{x}(t)] - f[x(t)] \frac{\delta f}{\delta x(t)}|_{x(t)=\hat{x}(t)} + \frac{(f[\hat{x}(t)] -...

I'm currently studying the Taylor series and I cannot figure out how the remainder term came to be. If anyone could clarify this for me, I would be really grateful ...! I understand that the Taylor series isn't always equal to f(x) for each x, so we put Rn at the end as the remainder term...

[URGENT] Taylor Series without using the built-in MATLAB "Taylor's Function" I have a MATLAB Test Tomorrow Please teach me the MATLAB programming to solve Taylor & Maclaurin Series, without using the built-in MATLAB "Taylor's Function" Please explain the procedure to solve them using the...

Homework Statement Find the taylor series of f(x)=1/(x)^(1/2) ; a=9 2. The attempt at a solution f(x) = (x)^(-1/2) f'(x) = -(1/2)*x^(-3/2) f''(x) = (1/2)*(3/2)*x^(-5/2) f'''(x) = -(1/2)*(3/2)*(5/2)*x^(-7/2) f''''(x) = (1/2)*(3/2)*(5/2)*(7/2)*x^(-11/2) f(9) =...

Trying to find the Taylor Series for cos(x) where x0 is PI. I've gotten cos(x) -1 -sin(x) 0 -cos(x) 1 sin(x) 0 cos(x) -1 It's clearly 0 every other term so I need 2k or 2k-1. But the -1 term switches between -1 and 1 How in world do I deal with this? xD Thanks for any...

The series is: (33/5) - (34/7) + (35/9) - (36/11)+... Looking at this, I'm guessing I can use the Taylor Series for arctan(x) but I don't know how to apply it or where to begin. Any help is greatly appreciated.

The Taylor Series of sin(x)=x-(x3/3!)+(x5/5!)-... What function of sin gives the following: (\pi2/(22) - (\pi4/(24*3!)+ (\pi6/(26*5!) - (\pi8/(28*7!)+... Please help me. Thank you.

Use taylor series method to compute the integral from 1 to 2 of [sin(x2)] / (x2) with 10-3 precision.

Homework Statement What function produces the following: (\pi2/(22)) - (\pi4/(24*3!)) + (\pi6/(26*5!)) - (\pi8/(28*7!)) I'm sure this is a sin function. But I can't figure out what exactly is the function. Please help.

Homework Statement Use taylor series method to compute the integral from 1 to 2 of [sin(x2)] / (x2) with 10 -3 precision Homework Equations The Attempt at a Solution I'm not sure where to start. Someone please help me.

Homework Statement For what values of x do you expect the following Taylor series to converge? sqrt(x^{2}-x-2) Homework Equations I'm not too sure The Attempt at a Solution Well quite frankly I have no idea what to do. If someone can push me in the right direction I'll get the rest done.

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)...

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...

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 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.

Hi, My lecture had gave a project about analyzing and discussion about - Taylor Series. I had done some research and tried understand and solve the question, but I'm in trouble now. I could only complete No.1 and No.2 (don't know whether is correct or not), I stuck at No.3 I have no idea...

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 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?

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 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 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 Q1) Use the Taylor series of f (x), centered at x0 to show that F1 =[ f (x + h) - f (x)]/h F2 =[ f (x) - f (x - h) ]/h F3 =[ f (x + h) - f (x - h) ]/2h F4 =[ f (x - 2h) - 8 f (x - h) + 8 f (x + h) - f (x + 2h) ]/12h are all estimates of f '(x). What is the error...

Homework Statement Determine the Taylor Series for f(x) = ln(1-3x) about x = 0Homework Equations ln(1+x) = \sum\fract(-1)^n^+^1 x^n /{n}The Attempt at a Solution ln(1-3x) = ln(1+(-3x)) ln(1+(-3x)) = \sum\fract(-1)^n^+^2 x^3^n /{n} Is that right?

Hi. How can I derive the Taylor series expansion and the radius of convergence for hyperbolic tangent tanh(x) around the point x=0. I can find the expression for the above in various sites, but the proof is'nt discussed. I guess the above question reduces to how can I get the expression...

I know that the Taylor Series of f(x)= \frac{1}{1+x^2} around x0 = 0 is 1 - x^2 + x^4 + ... + (-1)^n x^{2n} + ... for |x|<1 But what I want is to construct the Taylor Series of f(x)=...

Okay so suppose I have the Initial Value Problem: \left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{IVP} NB. I am considering only real functions of real variables. If f(x,y) is...