Taylor series Definition and 480 Threads

Can anyone please give me an example of a real function that is indefinitely derivable at some point x=a, and whose Taylor series centered around that point only converges at that point? I've searched and searched but I can't come up with an example:P Thank you:)

how to find the taylor series for y(x)=\sin^2 x i need to develop a general series which reaches to the n'th member so i can't keep doing derivatives on this function till the n'th member how to solve this??

Hey all, So I have a physics final coming up and I have been reviewing series. I realized that I'm not quite sure on what the differences are between a Taylor series and a power series. From what I think is true, a taylor series is essentially a specific type of power series. Would it be...

It is known that \sum\limits_{k = 0}^\infty {\frac{{N^k }}{{k!}}} = e^N I am looking for any asymptotic approximation which gives \sum\limits_{k = 0}^M {\frac{{N^k }}{{k!}}} = ? where M\leq N an integer. This is not an homework

How is it possible to see that exp(i\phi) is periodic with period 2\pi from the Taylor series? So basically it boils down to if is it easy to see that \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}(2\pi)^{2n}=1 ? Or any other suggestions?

Homework Statement Expand V(z + dz, t). I have seen problems like this in both my EnM and semiconductor courses but it's bothering me because I don't understand how the Taylor series is being used in this case... Homework Equations The Attempt at a Solution Taylor series...

Homework Statement Using Taylor expansion, show that the one-sided formula (f_-2-4f_-1+3f)/2h is indeed O(h2). Here f-2, for example, stands for f(xo-2h), and f-1 = f(xo-h), so on. The Attempt at a Solution Can some1 help me get starte, I am greatly confused

Homework Statement I need to find the following limit. Homework Equations \lim_{x\rightarrow0}\frac{(x-\sinh x)(\cosh x- \cos x)}{(5+\sin x \ln x) \sin^3 x (e^{x^2}-1)} The Attempt at a Solution I think it's got to be something with Taylor series, but I don't really know how to do it.

I really need some tips on taylor series...Im trying to learn it myself but i couldn't understand what's on the book... Can anyone who has learned this give me some tips...like what's the difference between it and power series (i know it's one kind of power series), why people develop it, and...

Questions: Is there a quicker way to find the formula for the nth derivative of a function, instead of finding the first several derivatives and trying to find a pattern, and using that pattern to form the equation for the nth derivative? Also, is there a formula for the nth derivative...

I was wondering if someone could help me with Goldstein's equation 6.3 (3rd Edition). It is the chapter of oscillations and all that he has done in the equation is to expand it in the form of a Taylor series. I can't seem to get how all those ni's come to get there.

Homework Statement Find the taylor series of \frac{1+z}{1-z} where z is a complex number and |z| < 1 Homework Equations \sum^{\infty}_{0} z^n = \frac{1}{1-z} if |z| < 1 The Attempt at a Solution \sum^{\infty}_{0} z^n = \frac{1}{1-z} \frac{1+z}{1-z} =...

Homework Statement 1/(4x-5) - z/(3x-2) based @ 0, answers are in those z things.. sigma Homework Equations i think we use sigma of e^x, but idk how... The Attempt at a Solution since tayor sereis of e^x is like 1/x, do i plug 4x-5 in? thanks

Homework Statement how to you find like the answer for f(1.5), or f(1.00001) those kind of question? thanks with like eq. = f(b)(x-b)... am i making sense? thanks

Homework Statement Need to calculate fractional uncertainty f, of M (mass of a star in this case), where f is much less than one. The hint i was given was all i need to know is M \alpha d3, and use a taylor expansion to the first order in f. M = mass of a star, d = distance to star...

not sure I get the Taylor Series... Hello Everyone. I understand that the taylor series approximate a function locally about a point, within the radius of convergence. If we use the Taylor series it means that we do not know the function itself. But to find the taylor series we need the...

Homework Statement find the first four nonzero terms in the power series expansion of tan(x) about a=0 Homework Equations \Sigma_{n=0}^{\infty} \frac{f^n (a)}{n!}(x-a)^n The Attempt at a Solution Well the series has a zero term at each even n (0,2,4 etc) for n=1 I got x, for...

Many of you have probably used the book Differential Equations by Lomen & Lovelock. For my class I'm working on Example 2, Page 153. You don't need to see the book, though, to help me out. It's a four-part problem and I'm on the last step not knowing where to take it. In Part B, we...

Homework Statement Let T_(4)(x): be the Taylor polynomial of degree 4 of the function ` f(x) = ln(1+x) ` at `a = 0 `. Suppose you approximate ` f(x) ` by ` T_(4)(x) `, find all positive values of x for which this approximation is within 0.001 of the right answer. (Hint: use the...

Homework Statement The Taylor series for f(x) = ln(sec(x)) at a = 0 is sum_(n=0to infinity) c(sub n) (x)^n. Find the first few coefficients. The Attempt at a Solution I've been trying to figure out where to start by looking it up...I've seen instructions that each coefficient is...

Homework Statement im being asked for the first 4 non zero values for the taylor expansion of exp(x) which is simple, but then it asks for the range of x values that are valid for the expansion. i have never come across ths before - any idea?

Hi am trying to solve this Taylor series with 3 variables but my result is not equal to the solution- So i think i might be wrong expanding the taylor series, or the solution is not correct Homework Statement Find an a approximated value for the function f(x,y,z) = 2x + ( 1 + y) * sin z at the...

Deduce that the Taylor series about 0 of 1/sqrt(1-4x) is the series summation (2n choose n) x^n. From this conclude that summation (2n choose n) x^n converges to 1/sqrt(1-4x) for x in (-1/4,1/4). Then show that summation (2n choose n) (-1/4)^n = 1/sqrt(1-4(-1/4)) = 1/sqrt(2) What I know...

Homework Statement Find the 3rd-order Maclaurin Polynomial (i.e. P3,o(u)) for the function f(u) = sin u, together with an upper bound on the magnitude of the associated error (as a function of u), if this is to be used as an approximation to f on the interval [0,2]. I did the question...

Homework Statement use an appropriate local quadratic approximation to approximate the square root of 36.03 Homework Equations not sure The Attempt at a Solution missed a day of class

Problem Statement Compute the Taylor Series expansion of f(x) = exp(-x^2) around 0 and use it to find an approximate value of the integral (from 0 to 0.1) of exp(-t^2) dt Solution Part1: First to compute the Taylor Series - I am pretty sure about this step so I will not give details...

When I tried to learn the Taylor series , I could not comprehend why a infinite series can represent a function Would anyone be kind enough to teach me the Taylor series? thank you:smile: PS. I am 18 , having the high school Math knowledge including Calculus

Homework Statement Find the Taylor series for f(x) = sin x centered at a = pi / 2 Homework Equations The Attempt at a Solution Taylor series is a new series for me. I believe the first step is to start taking the derivative of the Taylor series. f(x) = sinx f'(x) =...

Homework Statement Use the Taylor series about x = a to verify the second derivative test for a max or min. Show if f'(a) = 0 then f''(a) > 0 implies a min point at x = a ... Hint for a min point you must show that f(x) > f(a) for all x near enough to a. Homework Equations The Attempt at a...

I am trying to find the maclaurin series for f(x) = (1 + x)^(-3) --> what is the best way of doing this--to make a table and look for a trend in f^(n)?

Homework Statement Find the Maclaurin series of the function f(x) = 5(x^2)sin(5x)Homework Equations \sum(Cn*x^n) The Attempt at a Solution I'm supposed to enter in c3-c7 I already know that c4 and c6 are 0 because the derivative is something*sin(0)=0 but for the odd numbered c's I am having...

Homework Statement z is a complex number. find the taylor series expansion for g(z)=1/(z^3) about z0= 2.in what domain does the taylor series of g converge. z0 is z subscript 0 Homework Equations The Attempt at a Solution I wrote g(z)=1/(z^3) = 1/(2+(z^3)-2) = (1/2)*1/(1+(z^3...

[SOLVED] Taylor Series Question I have to find the Taylor series of \frac{3}{z-4i} about -5. Therefore, we want the series in powers of z+5. Now, following the textbook it appears that we want to get this in a form that resembles a geometric series so that we can easily express the Taylor...

I'm unclear on what they are asking in this homework problem. Suppose we know a function f(z) is analytic in the finite z plane apart from singularities at z = i and z=-1. Moreover, let f(z) be given by the Taylor series: f(z)=\displaystyle\sum_{j=0}^{\infty}a_{j}z^{j} where aj is...

Homework Statement Use the "Three Term" Taylor's approximation to find approximate values y_1 through y_20 with h=.1 for this Initial Value Problem: y'= cosh(4x^2-2y^2) y(0)=14 And write a computer program to do the grunt work approximation Homework Equations The Attempt...

How does one prove taylor series? Is it proven the same way as Maclaurin's Series(Which i know is a special case of taylor series) f(x)=A_0+A_1x+A_2x^2+A_3x^3+... f(\alpha)=A_0+A_1\alpha+A_2(\alpha)^2+A_3(\alpha)^3+... this kinda doesn't seem like a good way to prove it...as that is how I...

Homework Statement Consider f(x) = 1 + x + 2x^2+3x^3. Using Taylor series approxomation, approximate f(x) arround x=x0 and x=0 by a linear function Homework Equations The Attempt at a Solution This is the first time that I have seen Taylor series and I am totally lost on how to...

Homework Statement I've been asked to: Use the real Taylor series formulae e^{x} = 1 + x + O(x^{2}) cos x = 1 + O(x^{2}) sin x = x(1 + O(x^{2})) where O(x^{2}) means we are omitting terms proportional to power x^{2} (i.e., \lim_{x\rightarrow0} \frac{O(x^{2})}{x^{2}} = C where C is a...

I was going through the derivation of the Taylors series in my book (Engineering Mathematics by Jaggi & Mathur), and there was one step that escaped me. They proved that the derivative of f(x+h) is the same wrt h and wrt (x+h). If someone could explain that, Id be really grateful.

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...