Taylor series Definition and 492 Threads

Homework Statement Let f(z) be a function that is analytic for all |z|≤1, with the exception of z_0, which lies on the circle |z|=1. f(z) has a first order pole at z_0. Letting Ʃ a_n z^n be the Maclaurin expansion of the function, prove that z_0 = lim_(n→∞) a_n/a_(n+1) Homework Equations...

Homework Statement Find the Taylor series expansion of f(x) = (x-1)/(1+(x-1)^2) about x=1 and use this to compute f(9)(1) and f(10)(1) Homework Equations The sum from n=0 to infinity of f(k)(c)/(k!) (x-c)k The Attempt at a Solution I'm not sure how to approach this...

I am reading through a worked example of the Taylor series expansion of Sinh(z) about z=j*Pi The example states: sinh(j*Pi)=cos(Pi)*Sinh(0) +jcosh(x)sin(y) I am unsure of this relation. I understand why the x terms are zero but don't know the relation to expand sinh. Can anyone shed...

Homework Statement Find the Taylor series expansion for f(x)=x*e^(-x^2) about x = -1 Homework Equations The Attempt at a Solution I have tried replacing x with (x-1) and f(x-1) = (x-1)*e^(-(x-1)^2). Consider the power series for e^(-(x-1)^2) about x = 0, f(x-1) =...

Find the Taylor series expansions for f(x)=x*e^(-x^2) about x = -1 -(1/E) - (x + 1)/E + (x + 1)^2/E + (5 (x + 1)^3)/(3 E) + (x + 1)^4/( 6 E) - (23 (x + 1)^5)/(30 E) - (29 (x + 1)^6)/(90 E) + ( 103 (x + 1)^7)/(630 E)... This is the answer from Mathematica but i don't know how it goes. Can...

Homework Statement Find the sums of the following series: S1=1+(x^3)/(3!)+(x^6)/(6!)+... S2=x+(x^4)/(4!)+(x^7)/(7!)+... S3=(x^2)/(2!)+(x^5)/(5!)+(x^8)/(8!)+... Homework Equations Perhaps Taylor series? The Attempt at a Solution I spotted that adding S1+S2+S3=e^x, but I don't...

Homework Statement Find the Taylor series expansions for f(z) = −1/z^2 about z = i + 1. Homework Equations The Attempt at a Solution I'm just not sure what format I'm supposed to leave it in. Is it meant too look like this: f(z)=f(i+1)+f'(i+1)(x-i-1)... or this Ʃ\frac{1}{n!}f^{(n)}(1+i) *...

Hello! I am wondering if someone could let me know if my understanding is right or wrong. The Taylor series gives the function in the form of a sum of an infinite series. From this an approximation of the change in the function can be derived: f_{a} and f_{a,a} are the first and second...

The equation starts at B and this is my attempt. As you can see it soon complicates and doesn't look like what t should since I already know what the Taylor series of his function should look like. Is there some clever trick to it that I am missing? PS the series is centred around x = 0...

Homework Statement This problem arises from the following ODE: \epsilon y'' + y' + y = 0, y(0) = \alpha, y(1) = \beta where 0 < x < 1, 0 < \epsilon \ll 1 Find the exact solution and expand it in a Taylor series for small \epsilon Homework Equations I guess knowing the Taylor...

Show that, with an appropriate choice of constant c, the taylor series of (1+cx)ln(1+x) has terms which decay as 1/n^2 I know that ln(1+x) decays as 1/n, but I don't know how to show the above. Please help. Thanks in advance

I have to prove that \cos(x) = 1 - \frac{x^2}{2} + O(x^4) (x \to 0) My ugly attempt: \lim_{x \to 0} \frac{\cos(x) - 1 + \frac{x^2}{2}}{x^4} \lim_{x \to 0} \frac{\cos(x) - 1}{x^4} + \frac{1}{2x^2} \lim_{x \to 0} \frac{\sin(x)}{4x^3} + \frac{1}{2x^2} \lim_{x \to 0}...

Homework Statement Hello, I'm in the middle of solving for the Taylor series of the function: f(x)=sin(2x)ln(1-x) up to n = 4. The Attempt at a Solution So far, I've been strictly taking its derivatives until I reach the fourth. It's becoming a very long process considering it's...

Homework Statement For f(z) = 1/(1+z^2) a) find the taylor series centred at the origin and the radius of convergence. b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius. Homework...

Given a partial differential equation, how would one go about implementing the forward finite difference method to the Taylor series?

Homework Statement https://skydrive.live.com/?cid=6b041751c72e14ad#!/?cid=6b041751c72e14ad&sc=photos&uc=3&id=6B041751C72E14AD%21149!cid=6B041751C72E14AD&id=6B041751C72E14AD%21154&sc=photos The Attempt at a Solution...

Homework Statement I posted this already but decided to revive this thread since I re-worked the problem. Consider dy/dx=x+y, a function of both x and y subject to initial condition, y(x0)=y0. Use Taylor series to determine y(x0+\Deltax) to 4th order accuracy. Initial condition: x0=0...

Hi By some googling it seems like there exist some kind of expansion of the Taylor series for statistical functionals. I can however, not sort out how it is working and what the derivative-equivalent of the functional actually is. My situation is that I have a functional, say \theta which...

Homework Statement Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x) Homework Equations The Attempt at a Solution Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x) f(x) =...

New Question (Changed Old one) - Taylor Polynomial - Upper Bound for Absolute Error Homework Statement (a) Find the 3-rd degree Taylor polynomial of sin(pix) centered at x=1. (b) Use (a) to approximate sin(1.1*pi) (c) Use the remainder term to find an upper bound for the absolute error in...

Homework Statement Hi, I'm really struggling with trying to come up with the error bound when doing taylor series problems Use the reaminder term to estimate the absolute error in approximating the following quantitites with the nth-order Taylor Polynomial cnetered at 0. Estimates are...

hello, I'm examinating the theorem of power series, specially taylor series I know a function f(x) can be written as a series of polynomials. but using the taylor series it says that the convergence of that function is about a point a by using the Maclaurinseries a = 0 , so examinating...

So I'm computing a second order Taylor series expansion on a function that has multiple variables. So far I have this I(x,y,t)=dI/dx(change in x)+dI/dy(change in y)+dI/dt(change in t)+2nd order terms Would it still be a better approximation than just he first order if I included some...

Now although this is a problem for my EE course, it is more of a calculus question so I figured I would receive the best answers by posting it in this section. I have just started on the problem but could use some input on my thoughts. So here we go (there are two parts): (problem screenshot is...

When a Taylor Series is generated from a functions n derivatives at a single point, then is that series for any value of x equal to the original function for any value x ? For example graph the original function (x) from x= 0 to x = 10. Now plug into the Taylor Expansion for x , values...

Homework Statement series expansion at c=2 of ln(x^2+x-6) Homework Equations The Attempt at a Solution After substituting y= x -2 we get ln(y^2+5y) = ln(y) + ln(y+5) but I am not kinda sure how to use the taylor series of ln(1+x)...

I read wikipedia article also but I can't find the proof of taylor series and from where it came from??

Usually to do the remainder we take Rn(x) = (f differentiated n+1 times at a ).(x-c)n+1/(n+1)!, but when my function is sin(x) do i take (f differentiated 2n+2 times at a ).(x-c)2n+2/(2n+2)!? Thanks

Homework Statement How many terms of the taylor series of the cosine function about c = 0 are needed to calculate cosine 2 to an accuracy of 1 / 10000 The Attempt at a Solution I have said that |Rn(2)| = |cosn+1(a) 2n+1/(n+1)!|<2n+1/(n+1!) Now i can't do it ...

I consider an array of lattice points and construct a vector at each lattice points. How to convert this discrete system into a continuum one by using the Taylor series expansion by considering the lattice distance say \lambda? thanks in well advance?

in general I'm trying to figure out a way to work with taylor series more efficiently. this means i want to be able to write down the taylor series of a complicated function just by knowing the taylor series(es?) of the component functions. I've figured out how to do products and quotients...

How do I use Taylor Series to show f(P) is a local maximum at a stationary point P if the Hessian matrix is negative definite. I understand that some of the coefficients of the terms of the taylor series expansion are the coordinates of the Hessian matrix but for the f_xy term there is no...

Solved Thanks guys

Homework Statement Let v(λ) = ∫0∞ λe-λy f(y)dy a) Prove that limλ→ ∞ v(λ) = f(0). b) Determine and prove limλ→ ∞ λ(v(λ)- f(0)). α β γ δ ε ζ η θ ι κ λ μ ν ξ ο ° π ρ ς σ τ υ φ χ ψ ω Ω ~ ≈ ≠ ≡ ± ≤ ≥ Δ ∇ Σ ∂ ∫ ∏ → ∞ Homework Equations Assume there exists some constant M where maxx∈ℝ...

Homework Statement Find the Taylor Series of 1/x centered at c = 1. Homework Equations \sum_{n=0}^{\infty} f^n (c) \frac{(x-c)^n}{n!} The Attempt at a Solution I made a list of the derivatives: f(x) = 1/x f'(x) = -1/x2 f''(x) = 2/x3 f'''(x) = -6/x4 f(1) = 1 f'(1) =...

Homework Statement a. Find the first four nonzero terms in the Taylor series expansion about x = 0 for f(x) = (1+x)^.5 b. Use the results found in part (a) to find the first four nonzero terms in the Taylor series expansion about x= 0 for g(x) = (1 + x^3)^.5 c. Find the first four...

Homework Statement The Taylor series of function f(x)=ln(x) at a=7 is given by: f(x)=\sum^{\infty}_{n=0}c_{n}(x-7)^{n} Determine the interval of convergence The Attempt at a Solution I have worked out that the series would be of the form...

Do you just replace the x's with (x-3)'s? Since e^(-x^2) is defined as the taylor series though, it seems like the answer should be the same as the series about x=0. Thanks! P.S. does anyone know how to resize images? :$

Homework Statement The function f(x)=ln(10-x) is represented as a power series: \sum^{\infty}_{n=0}a_{n}x^{n} Find the first few coefficients in the power series. Hint: First find the power series for the derivative of . The Attempt at a Solution Okay, start seems fairly...

I have a couple of general questions, combined with this one specific question Homework Statement Find the Taylor or MacLauren series centered about the given value for the following function, determine the radius of convergence Homework Equations \mathrm{Ln}\ z, 2 The Attempt at a Solution...

Homework Statement find the taylor series of ln(1+x) centered at zero Homework Equations from 0 to infinity ∑ cn(x-a)n cn = f(n)(a)/n! The Attempt at a Solution f(x) = ln(1+x) f'(x) = 1/(1+x) f''(x) = -1/(1+x)2 f'''(x) = 2/(1+x)3 f''''(x) = -6/(1+x)4 f(0) = 0...

Homework Statement Find the Taylor series for f(x) centered at the given value of a. (Assume that f has a power series expansion. Do not show that Rn(x)--> 0.) f(x) = x^3, a = -1Homework Equations f(x) = f(a)+f'(a)(x-a)+(f''(a)/2!)(x-a)^2+(f'''(a)/3!)(x-a)^3+...+(f(nth...

Homework Statement Expand f(z) = (3z+1)/(15+2z-z^{2}) at z=1 and find the circle of convergence. Homework Equations The Attempt at a Solution I think this is pretty straight forward, but I want to make sure I'm doing everything correctly. I used a power series...

Homework Statement [PLAIN]http://img822.imageshack.us/img822/427/scangj.jpg Homework Equations The Attempt at a Solution Hi, could anyone help me with part b of this question, part a I have completed, however I seem to be drawing a blank on the second part

Homework Statement Suppose that: sum [a_n (n-1)^n] is the Talyor series representation of tanh(z) at the point z = 1. What is the largest subset of the complex plane such that this series converges? Note: 'sum' represents the sum from n=0 to infinity Homework Equations tanh(z) =...

Homework Statement Expand the function f(E) as a Taylor series.Homework Equations f(E)=E/(KT)+(Ec/E)1/2 The Attempt at a Solution E=Eo So it says that F(E)~Ao+A1(E-Eo)+A2(E-Eo)2... I need to find out what Ao A1 and A2 are, but not sure how to do that. It says as a hint that A1=0 becasue f(E)...

Background: I'm trying to transform the gaussian distribution from flat space to curved space. I start with the flat, 1D gaussian distribution in the form \[{\textstyle{1 \over {{{(\pi {\Delta ^2})}^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em...

Homework Statement approximate sin13 by using the Taylor series using the TI84. Add for n=150 Homework Equations the infinite sums for sinx is ((-1)^n)(x^(2n+1)/(2n+1)!) The Attempt at a Solution I'm new to programming so i don't have any idea on where to start. I was...

how would you use Taylor Series to answer this: Find a value of h such that for |x|<h implies sin(x)=x-x^3/6 +x^5/120 + R where |R|< 10^-4?

find the first three non zero terms in the Taylor Series about z=0 of exp(z sin z) i have little idea how to even start on the question because it is exp to the power of z sin z and it just looks too complicated. i hav tried looking thru txtbooks for something similar but no similar question...