Taylor series Definition and 492 Threads

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 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...

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...

TASK: Assuming a complex function f(z) can be expanded as a Taylor series around z=0, i.e.: f(z)=\sum_{n=0}^{\infty}a_{n}z^n Setting z=r*exp(i*theta), assuming a_n is real, find real part u(r, theta), imaginary part v(r,theta). Comment the result, especially for r=1. MY SOLUTION...

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 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...

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 +...

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...

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?

Compute the Taylor series for f(x)= sq root (x) about x=1. Determine where the series sconverges absolutely, converges conditionally, and diverges. Hint: 2(k!)=2*4*6...(2k-2)*2k. Also 1<2, 3<4, 5<6,..., 2k-1<2k should help you out with a comparision.

hi, I'm wondering if someone can help me out with this question: "What are the first two non-zero terms of the Taylor series f(z) = \frac {sin(z)} {1 - z^4} expanded about z = 0. (Don't use any differentiation. Just cross multiply and do mental arithmetic)" I know the formula for...

may i know from ln(1-x) how to become - [infinity (sum) k=1] x^k / k ? pls help

Dear friends, I have a question on a taylor series, that is this one: A·e^(i (x)) That is: cos (x)+ i sin (x) becouse of the taylor's. But, is this wrong? A·e^(v (x)) = cos (x)+ v sin (x) (v is a vector). Tks.

Hi all, here's the problem: given: tan^(-1)= x - x^3/3 + x^5/5 using the result tan^(-1) (1)= pi/4 how many terms of the series are needed to calculate pi to ten places of decimals? note: this is supposed to say tan^(-1) and tan^(-1)[1] respectively Does anyone know whether...

y=kcosh(x/k) = k(e^(x/w) + e^(-x/k)) y ~ k[1 + x/k + (1/2!)(x^2/k^2) + (1/3!)(x^3/k^3)] +...+ k[1 - x/k +(1/2!)(x^2/k^2) - (1/3!)(x^3/k^3) +...] All odd terms except 1 cancel out. So we are left with y = k [2 + (2/2!)(x^2/k^2) + (2/4!)(x^4/k^4) + (2/6!)(x^6/k^6) +...] I've been...

Is there a way to get the Taylor series of 1/sqrt(cosx), without using the direct f(x)=f(0)+xf'(0)+(x^2/2!)f''(0)+(x^3/3!)f'''(0)... form, just by manipulating it if you already know the series for cosx?

Hi, I was reading this math book once... and it had a method for solving differential equations of 1st (And maybe 2nd? I don't remember) order by using simple Taylor series... I didn't even have to understand much of what was going on, except that I followed some simple rule and I ended up...

Hi, can someone explain me the relation between the degree of a taylor series (TS) and the error. It is for my class of numerical method, and I do not find a response to my question in my textbook. I mean when we have a function Q with two variables x and y,and we use a version of TS to...

Hi, can someone explain me the relation between the degree of a taylor series (TS) and the error. It is for my class of numerical method, and I do not find a response to my question in my textbook. I mean when we have a function Q with two variables x and y,and we use a version of TS to...

Hi , I have some difficulties to solve this problem. It is from my numerical methods class but the problem is about taylor series: It is known that for 4 < x < 6, the absolute value of the m-th derivative of a certain function f(x) is bounded by m times the absolute value of the quadratic...

Hi, I have a question about Taylor series: I know that for a function f(x), you can expand it about a point x=a, which is given by: f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ... but I would like to do it for f(x+a) instead of f(x), and expand it about the very same point...

I need to find the first three terms of this series. Am I correct in saying z^i = exp(i*Log(z)), then using the taylor series for e^z, giving me: (i*Log(z)) - 1/2*(Log(z))^2 - i/6*(Log(z))^3 + 1/24*(Log(z))^4 + ... I haven't worked it out, but this seems to mean that the coefficients...

How do i show that B_{x}(x+dx,y,z)-B_{x}(x,y,z)\approx \frac{\partial B_{x}(x,y,z)}{\partial x} dx using a Taylor series to the first term. Using a Taylor series does B(x) = B(a) + B'(a)(x-a)? In that case what would B(x+dx) be and how can i obtain the desired result from this? Thanks in...

I was wondering if someone can give me some tips for finding the taylor series of functions. For example this was a test question we had: Find the taylor series of f(x)=ln(x) about x=e I know how to start it off but I get confused halfway through and can't seem to figure out what to do...

I understand what a linear approximation, and how it is derived using the point-slope formula: f(x)\approx f(a)+f'(a)(x-a) These are the first three terms of a Taylor series, so I was wondering how the rest was derived? Thanks for your help.

Problem Find the sum of the series s(x) = \sum _{n=1} ^{\infty} \frac{1}{2^{n}} \tan \frac{x}{2^n} Solution If s(x) = \sum _{n=1} ^{\infty} \frac{1}{2^{n}} \tan \frac{x}{2^n} = \frac{x}{3} + \frac{x^3}{45} + \frac{2x^5}{945} + \dotsb \cot x = \frac{1}{x} - \frac{x}{3} -...

Help me out with this Taylor series problem: The Taylor series for sin x about x = 0 is x-x^3/3!+x^5/5!-... If f is a function such that f '(x)=sin(x^2), then the coefficient of x^7 in the Taylor series for f(x) about x=0 is? thanks

Taylor Series in x-a Hi, I've got a question about the use of dummy variables in Taylor Series. We are asked to expand: g(x) = xlnx In terms of (x-2). So originally, I used a dummy variable approach to try and find an answer. Let t = x-2, so x = t+2. g(x) = (t+2)ln(2+t)...

This problem has been bugging me and I can't seem to figure it out: y'' = e^y where y(0)= 0 and y'(0)= -1 I'm supposed to get the first 6 nonzero terms I know the form is: y(x) = y(0) + y'(0)x/1! + y''(0)x^2/2! + y'''(0)x^3/3! +... and I know the first two terms are y(x) = 0 -...

Find the 4th term of the Taylor series centerd at x=1 for f(x)=ln(x+1) f(x)=ln(1=x) f'(x)=(1+x)^-1 f"(x)=(-1)[(1+x)^-2] f"'(x)=(2)[(1+x)^-3] f""(x)=-6[(1+x)^-4)] Plug in 1: .6931 .5 -.25 .25 -.375 What do I do next? (Also, is the 4th term the 4th term starting with f(x)? or the...

We were gievn a question in tutorial last week asking us to calculate the Taylor series of the function f(x,y) = e^(x^(2) + y^(2)) to second order in h and k about the point x=0, y=0 I've got the forumla here with all the h's and k's in it and have it written down, but it's actually how to...

I'm having some problems expanding i^i, could anyone help? I know it becomes a real number somehow, and I'm familiar with the e^(i * pi) expansion, but is the i^i done in the same way?