Taylor Definition and 849 Threads

Homework Statement Let f(x) = x3ln(1+x2), and let the summation (from n=0 to infinity) anxn be the Taylor series for f about 0. Then what is a3? Homework Equations The Attempt at a Solution What?! I definitely don't expect the answer, but does anyone know how I could go about...

Homework Statement Let f(x)=x2 +3x -5, and let the summation (from n=0 to infinity) an (x-4)n be the Taylor series of f about 4. Find the values of a0, a1, a2, a3, and a4. Homework Equations The Attempt at a Solution What am I supposed to do with the summation? And what does it mean...

Homework Statement use the third degree Taylor polynomial of cos at 0 to show that the solutions of x2=cos x are approx. \pm\sqrt{2/3}, and find bounds on the error. Homework Equations P2n,0(x) = 1-x2/2!+x4/4!+...+(-1)nx2n/(2n)! The Attempt at a Solution when it says "third...

Homework Statement find Taylor polynomial for ln x of degree n, at 2 (Pn,2(x)) Homework Equations Pn,1(x)= (x-1) - (x-1)2/2 + ... + (-1)n-1(x-1)n/n The Attempt at a Solution there doesn't seem to be an obvious pattern to this. the coefficients for n=1 to n=4 are 2, -8, 24, -64...

I have a shaky understanding of problems concerning Taylor Series. For example, the question below. Let f(x)=\tan^{-1}\left(\frac{1+x}{1-x}\right) where -\frac{1}{2}\leq x \leq \frac{1}{2}. Find the value of f^{2005}(0) the Taylor Series of \tan^{-1} is...

The formula given by my instructor for a Taylor Series approximation of the second order at point (a,b) is f(a,b) + grad(f(a,b))x + 1/2 H(f(a,b)) x If you recognize this formula, do you know what the x vector is? Note: x is the x-vector, and H represents the Hessian Matrix. Thanks! The...

Homework Statement Find the Taylor Polynomial of degree 3 for the function f(x) = ln3x about a = 1/3Homework Equations NoneThe Attempt at a Solution I have found up to the fourth derivative of f(x) along with the values of the derivatives at x = 1/3. At this point i get Σ{(-1)kk!fk(1/3)}, but...

Hi! I am a 16 year old trying to figure out the application of taylor series. I understand most of its uses when applied to functions like e^x, sinx, cosx, but in a mechanics book, i am required to find delta-F, a finite change in a function F. Ostensibly, this appears to be a step that needs...

Homework Statement It seems that I'm a little bit lost about this exercise. It says: Find the taylors polynomial of third degree centered at the origin for z=\cos y \sin x. Estimate the error for: \Delta x=-0.15,\Delta y=0.2. So, I did the first part (the easy one), the taylors polynomial for...

When it says "about a point x=a", what does this mean? why not just say at x = a? Thanks

Homework Statement Obtain the Taylor series in powers of x + 1 for f(x) = x/(2 + x), giving the general term. Homework Equations The Attempt at a Solution Wrote it out as x*(1/1-(-(x+1)).

Homework Statement This is actually not a problem, it's something in my notes. The function I am supposed to be approximating is V(x) = V0(1 - ex/a)2 - V0 V0 and a are constants. Homework Equations The Attempt at a Solution It says that the function given is not a parabola. But it can be...

Hello, I am trying to evaluate the series \sum{\frac{x^n}{n!}e^{cn^2}} where c is a constant. I think this problem is equivalent to find f(x) such that \frac{d^{n}f(0)}{dx^{n}} = \frac{e^{cn^{2}}}{n!} I believe this must be a modified exponential since for c=0, it reduces to...

Homework Statement See figure attached. Homework Equations The Attempt at a Solution Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the, \frac{1}{x^{2}} \int x^{-2} = \frac{-1}{x} + C How do I...

So this was a textbook problem my professor did in lecture. I felt like I followed along with the logic as she went along, but after a few days and looking back it, I can't seem to recreate it genuinely. Homework Statement A ball is thrown with initial speed v0 up an inclined plane. The...

Hello, if I understand correctly the Taylor approximation for a=0 gives me the possibility to approximate a function, say sin(x), at any x. But, what gives me Taylor polynomial at some point http://latex.codecogs.com/gif.latex?a\neq0 ,[/URL] what's the difference? what does it mean centred...

hello, please help to calculate the taylor polynomial for http://latex.codecogs.com/gif.latex?f(x)=x^{x}-1 around the point a=1 i thought to write it as g(x)=x^x and then f(x)=g(x)-1 and then find the polynomial for g(x) as lng(x)=xln(x) but it seems incorrect.

I am trying to linearize a function, f(x), where x is a normally distributed N(0,1) random variable. How can I perform a taylor series expansion around a deterministic value x0? Thanks.

Homework Statement The velocity of a proton relative to our galaxy is vp/c = 1-(0.5*10^20), i.e. almost one. Such protons are actually observed. When velocity it very nearly one \gamma is very large. 1/\gamma is very small. Use Taylor series to show that for v almost one we have...

I am wroking through an electrodynamics textbook and there is this Taylor expansion to do later a multipole expansion. But I can't figure out how the author does it. Please any help? the expansion: \frac{1}{|\vec{r}-\vec{r'}|} = \frac{1}{r} - \sum^3_{i=1} x'_i \frac{\partial}{\partial...

Can anyone help me for the leading order terms in the taylor series for the function f(x,y) = Sqrt(a*x^8+b*x^4*y^4+y^8), centered at x=0,y = 0 and a,b,c constants?

where do a multiple Taylor series converge ?? i mean if given a function f(x,y) can i expand this f into a double Taylor series that will converge on a rectangle ? for example , if one can ensure that it converges for |x| <1 and |y| <1

Hello all, I am currently studying multivariable calculus, and I am interested in the Taylor series for two variable function. I am not sure where to begin; I cannot understand any of the proofs (which are apparently sparse) on the internet; they all just state it using a sigma sum; not...

I have a couple of questions about exercise **103 (yes, a two-star problem!) in Taylor & Wheeler's Spacetime Physics. In part (a), it says "For an atom \beta_r \leq Z/137 (Ex. 101), and for small Z, \beta_r \ll 1. Therefore \tan(d\phi) \approx d\phi \approx -\beta_r^2 \sin(\alpha)." But it's...

Hi, We need a generic expression of a taylor series nth term to find out the radius of convergence of the series. However, there are series where I don't think it is even possible to find a generic term. How do we find the radius of convergence in such cases? e.g. sqrt (1 - x^2) There...

"Partial" Taylor Series Expansion It has been awhile since I have had to use a Taylor series expansion (from scratch). I looked it up on wiki and the rules are easy enough, I am just a little confused as to how I apply it to a multivariable function, but only expand it about one variable...

I'm doing some review over summer before starting college, and one of the practice exams has a question pertaining to the remainder of a taylor series Homework Statement Show that \left|\cos{(1+x)}-\{\cos{(1)}(1-\frac{x^2}{2})-\sin{(1)}(x-\frac{x^3}{3!})\}\right|<\frac{1}{15000} for |x|<0.2...

Taylor Series Expansion About the Point "i" Homework Statement Calculate the radius of convergence of the Taylor series for \frac{1}{z^2-2z+2} about the point i. The Attempt at a Solution I can find the radius of convergence if I can determine the expansion but I can't seem to...

i can't understand how the got this variation of taylor series formula f(x+h)=\sum_{k=0}^{\infty}\frac{f^{(k)}(x)}{k!}(h)^k http://mathworld.wolfram.com/TaylorSeries.html when around some point there is no x-x_0

Homework Statement From the taylor series we can replace x =x_{0} + h but how does \delta f = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0}) become \delta f = hf(x_{0}, y_{0}) + kf(x_{0}, y_{0}) I can see the first step, but how do you get it to the second?Homework Equations The Attempt at a Solution

Hi, I'm doing calc-2, and I have hard time understanding and visualizing the idea of Taylor approximation in my head. By the same time I have no problems solving homework on this topic. Can someone please explain how I should visualize and think about approximations using Taylor Polynomials...

f(x)=sinx taylor series centered at pi/2 sum((-1)^n (x-pi/2)^(2n)/(2n)! , n=0,infty ) with radius of convergence infty

f(x)=1/(1-x^2)^(1/2) 1/x^(1/2)=1+ sum(( (-1)^n 1*3*5*7...(2n-1)(x-1)^n )/(2^n n! ) , n=1, infty ) thus 1/(1-x^2)^(1/2) = 1+ sum(( 1*3*5*7...(2n-1)(x^2)^n )/(2^n n! ) , n=1, infty ) is this a correct taylor series representation centered at 1

I was just pondering today how the kinematic equation for position looks like a taylor expansion. x = x0 + dx/dt *t + (1/2)*d2x/dt2*t2 I believe it continues like that, exactly like a taylor expansion does, so the next term would be (1/6)*d3x/dt3*t3 If it is indeed a taylor expansion, what...

Hello, I was wondering if anyone could explain to me the thought process behind how you find the maximum remainder of a Taylor series? I read the wiki article and didn't help me at all, http://en.wikipedia.org/wiki/Taylor's_theorem My book talks about something like this(image is...

Homework Statement For f(x) = xln(x), find the taylor series expansion of f(x) about x = 1, and write the infinite series in compact form. 2. The attempt at a solution I can find the expansion itself fine, these are the first few terms: 0 + (x-1) + \frac{(x-1)^{2}}{2!} -...

Homework Statement a) Using a geometric series, find the Taylor expansion of the function f(x) = x/(1+x^2) b) Use the series found in (a) to obtain the Taylor expansion of ln(1 + x^2) Homework Equations The Attempt at a Solution I really don't know where to start; I can't find...

Hi everyone. The problem I have to face is to perform a taylor series expansion of the integral \int_{-\infty}^{\infty}\frac{e^{-\sum_{i}\frac{x_{i}^{2}}{2\epsilon}}}{\sqrt{2\pi\epsilon}^{N}}\cdot e^{f(\{x\})}dx_{i}\ldots dx_{N} with respect to variance \epsilon. I find some difficulties...

so F = mgR2/(R+h)2 where R is the radius of the earth. consider the situation where h is much smaller than R. a) show that F is approximately equal to mg b)express F as mg multiplied by a series in h/R so i need help on getting started. would showing that F is approximately equal...

Homework Statement find the taylor series for the function f(x) = \frac{x^2+1}{4x+5} Homework Equations N/A The Attempt at a Solution how to do this? 1st attempt. i did turn it this term \frac{x}{4} + \frac{-5x+4}{16x+20} can i turn this to taylor series? maybe i know how to make...

I don't have anyone else to ask. So I have to ask you guys. I learned about Taylor series, and then I went back and looked at linear and quadratic approximations, and they are Taylor series except only taken so far. I'm pretty much just looking for confirmation on my idea, it seems perfect.

Homework Statement find the taylor polynomial f4 for f(x)=sin(2x) and a=pi/4 Homework Equations sin(x)=((-1)^n)(x^(2n+1))/((2n+1)!) The Attempt at a Solution so replace x with 2x? you get ((-1)^n)(2x)^(2n+1)/(2n+1)!) is this right?

Homework Statement find taylor series for \frac{x-1}{1+x} at x=1 Homework Equations The Attempt at a Solution how to change this form \frac{x-1}{1+x} to something like this \frac{1}{1+a} or \frac{1}{1-a} help me please T_T or should i do like this \sum\frac{f^n(1)(x-1)^n}{n!} and find...

Homework Statement Find the Taylor Series for f(w) = 1/w centered at w0 = 1 using 1/w = (1/1 + (w-1)). Show that the series converges when |w-1| < 1 Homework Equations use 1/w = (1/1 + (w-1)) The Attempt at a Solution

I have two equations: \ddot{x}^\mu + \ddot{y}^\mu + \Gamma^\mu{}_{\nu \lambda} (x+y)(\dot{x}^\nu+\dot{y}^\nu)(\dot{x}^\lambda+\dot{y}^\lambda)=0 and \ddot{x}^\mu + \Gamma^\mu{}_{\nu\lambda}(x) \dot{x}^\nu \dot{x}^\lambda=0 apparently if i taylor expand the first equation to first order...

Homework Statement Use the taylor's expansion of f(x)= x1/4 about x= 16 to estimate (16.1)1/4 Homework Equations Taylors formula: f(a) + f'(a) (x-a) + (f''(a)/2!) (x-a)2+...The Attempt at a Solution Ok I have calculate the taylor expansion to be: 2 + (1/32) (x-16)-(3/320) (x-16)2+ (7/262144)...

Homework Statement Derive a method for approximating f'''(x0) whose error term is of order h^{2} by expanding the function f in a fourth taylor polynomial about x0 and evaluating at x_{0} \pm h and x_{0} \pm 2h. Homework Equations The Attempt at a Solution I'm not sure where to...

Homework Statement This is a three part question: It is based off the first two sections. I'm pretty sure the first two answers are correct, but I have no idea how to do the third question. Write the First three nonzero terms and the general term of the Taylor series expansion about x=0...

Homework Statement If \sum_{n=0}^{\infty} a_{n}x^n is a Taylor series that converges to f(x) for all real x, then f'(1) = ? Homework Equations A Taylor series: \sum_{n=0}^{\infty} \frac {f^{(n)}(c)}{n!}(x-c)^n and the dirv of a Taylor series: f'(x)=\sum_{n=0}^{\infty}...

Homework Statement The question asks me to write out a taylor polynomial for 1/(1-x^2) of degree 2n+1 at 0. The Attempt at a Solution My answer was 1 + x^2 + x^4 + x^6 + ... + (x^4)/(1-x^2) which I just got from using hte geometric series formula. The textbook answer however...