Taylor Definition and 878 Threads

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...

In the Acknowledgements of "Exploring Black Holes" by Taylor and Wheeler they mention that a solutions manual was created by G.P. Sastry and several students. Does anyone know whether it is possible to get a hold of this solutions manual for us self learners. It would be a great help...

Taylor formula. Need help! we have f(a+h), where a-is a point, and h is a very small term, h->0. And we have the formula to evaluate the function y=f(x), around the point a, which is f(a+h)=f(a)+f'(a)h+o(h) --------(*) however when we want to take in consideration o(h) this formula does...

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...

Homework Statement For g=Hf = sin (f), use a Taylor expansion to determine the range of input for which the operator is approximately linear within 10 % Homework Equations The taylor series from 0 to 1 , the linearization, is the most appropriate equation The Attempt at a Solution...

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...

1) Let f(x) = (x^3) [cos(x^2)]. a) Find P_(4n+3) (x) (the 4n + 3-rd Taylor polynomial of f(x) ) b) Find f^(n) (0) for all natural numbers n. (the n-th derivative of f evaluated at 0) I know the definition of Taylor polynomial but I am still unable to do this quesiton. I tried to find the...

Homework Statement Linearize the system operator illustrated below by applying a Taylor series expansion. f(t) ----> e^f(t) -----> g(t) Homework Equations I only find the general form of a taylor series relevant. g(x)= sum (0,infinity) of [f^n*(a)*(x-a)^n]/n! The system is...

Homework Statement I'm trying to make the nth degree taylor polynomial for f(x)=sqrtx centered at 4 and then approximate sqrt(4.1) using the 5th degree polynomial I know that the polynomials are found using the form: P(x)= f(x)+f'(x)x+f''(x)x^2/2factorial...f^n(x)x^n/nfactorial so...

This is just part of a larger problem, but I have a basic equation r'=k-g*r, where k and a start out as constants, but then I need to treat everything as if it can vary slightly from the average. For this, I set r=r_ave+dr, g=g_ave+dg, and k=k_ave+dk. Now I need to work these into the first...

Ok there's something I don't get. I know for instance that the linear polynomial for say f = 91 + 2x + 3y + 8z + Quadratic(x, y, z) + Cubic(x, y, z) ... is 91 + 2x + 3y + 8z if the base point is (0, 0, 0). This is pretty clear. What I don't get is why when you take the base point to be say (1...

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 I wonder if there is a simpler way to obtain the first three non-zero terms of Taylor Expansion for the function \frac{Ln(1+x)}{1-x} about x=0? I differentiated it directly, but it was such a nightmare to do:mad: . So I am wondering if there is a simpler way to do it?

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...

Homework Statement I have the following question to answer: Show that (X^2/h^2)*((1/2*y1) - y2 + (1/2*y3)) + (X/h)*((-1/2 y1)+(1/2 y3))+y2 (sorry about the format) is equal to (taylor expansion): y = y2+(x(dy/dx)¦0 + (x^2/2*((d^2)y)/(dx^2))¦0 Homework Equations also given in...

any insight to this question? .. i mean.. usually people just do up to order 2.. find the taylor polynomial of order 3 based at (x, y) = (0, 0) for the function f(x, y) = (e^(x-2y)) / (1 + x^2 - y) how large do you have to take k so that the kth order taylor polynomial f about (0, 0)...

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 +...

Hey Everyone. I'm ALMOST finished this problem... To spare you the long story, I need to take the difference between an gravitational acceleration, and the same gravitational acceleration at a slightly larger height. The two functions are a(r) and a(r+d), where d is very small Now... VERY...

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...

In textbooks these polynomials are not normally presented as an infinite series (the single variables are). What is the reason for this and are they equally allowed to be in infinite series form hence infinite order just like the single variable Taylor Polynomials? Or are there more issues about...

Hi Guys, I have an assigment which I would very much appreciate if You would tell if I have done it correct :) Use the Taylor Polynomial for f(x) = \sqrt(x) of degree 2 in x = 100. To the the approximation for the value \sqrt(99) First I find the Taylor polynomial of degree 2...

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 Given a function f(x) = sqrt(x) is the Taylor Polynomial of degree 2 for that function: \frac{x^2}{2} - 99x + 4901 where x = 100 ? Sincerely Fred

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...

I'm having trouble determining the order of the pole of [exp(iz) - 1]/((z^2) + 4) at z=2i I know I can't just expand the exponential as 1 + iz + [(iz)^2]/2 ... because this formula only works near the origin. Can I still use Taylor's theorem to find the expansion at z=2i (i.e does...

(a) I found the answer to be: 1/(1-x) = 1 + x + x^2 + x^3 + ... + [x^(n+1)]/(1-x) for x != 1 *Note: "^" precedes a superscript, "!=" means "does not equal" (b) Use part (a) to find a Taylor polynomial of a general (3n)th degree for: f(x) = (1/x)*Integral[(1/(1 + t^3), t, 0, x] *Note...

the problem reads develop expansion of ln(1+z) of course I just tried throwing it into the formula for taylor expansions, however I do not know what F(a) is, the problem doesn't specify, so how can I use a taylor series?

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

I have had this book for a while and never really looked into it. It claims to be an easy/nonmathematical approach to relativity. Has anyone read this book before? Can I really understand what the subject matter is covering without any post-calculus math? Is it also a good beginer's guide to...

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.

Taylor rule of thumb?? When calculating limits by using taylor series is there any easy way to know how many elements that should be included in the taylor series? if I have \lim_{x\rightarrow\zero} \frac{exp(x-x^2)-Cos2x-Ln(1+x+2x^2)}{x^3} How do I know many terms to include in...

Can anyone please give me a hint on any of the following Taylor expansions? That would be so helpful! for all three: Find the first non-zero term in the Taylor series about x = 0 problem 1 \frac{1} {sin^2x} - \frac{1} {x^2} everytime I differentiate the result is zero...so that...

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...

So I'm studying Taylor Series (I work ahead of my calc class so that when we cover topics I already know them and they are easier to study..) and tonight I found a formula for taylor series and maclaurin series, and i used them to prove eulers identity. However, I don't really know much about...

How can you invert a Taylor serie? x=y+Ay^2+By^3+Cy^4... to y=ax+bx^2+cx^3 ... without the lagrange theorem... must go from x=y+Ay^2+By^3+Cy^4... to y=ax+bx^2+cx^3 ... Need help thanks!

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 have found the following TP (n=4) for g(x) = (1+5x)^1/5 P4(x) = 1+x-2x^2+6x^3-21x^4 Then they ask me to show that 0<E4(x)<80x^5 when x>0. I don't know how to start, or exactly what I am supposed to show...? I have found E4(x) to be( 399/[5(1+5X)^24/5] ) *x^5... And 0<X<x ...?

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...

Find the thrid taylor polynomial P3(x) for the function f(x) = \sqrt{x+1} about a=0. Approximate f(0.5) using P3(x) and find actual error thus Maclaurin series f(x) = f(0) + f'(0)x + \frac{f''(0)}{2} x^2 + \frac{f^{3}(0)}{6} x^3 f(x) = x + \frac{1}{2} x - \frac{1}{8} x^2 +...

I am supposed to find an approximation of this integral evaluated between the limits 0 and 1 using a taylor expansion for cos x: \int \frac{1 - cos x}{x}dx and given cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}... i should get a simple series similar to this for...

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...