Taylor Definition and 878 Threads

Homework Statement Evaluate the anti derivative ∫e^x^2 dx as a Taylor Series Homework Equations \frac{f^(n)(a)}{n!}(x-a)^n The Attempt at a Solution Where do I start, I am not sure I understand the question

Homework Statement I read that the taylor series was a way to approximate the a function f(x) graphically, by addition and subtraction. So say I have \frac{1}{1-x}=1+x+x^{2}+x^{3}+...+x^{n}... suppose x=3, then the left and right side of the equation can't possibly equal the same thing...

From page 54 of 'Modern Quantum Mechanics, revised edition" by J. J. Sakurai. Obtaining equation (1.7.15), \begin{eqnarray} \left(1- \frac{ip\Delta x'}{\hbar} \right) \mid \alpha \rangle & = & \int dx' \mathcal{T} ( \Delta x' ) \mid x' \rangle \langle x' \mid \alpha \rangle \\ & = & \int...

In Taylor's advanced text on classical mechanics, he gives an example of a non-holonomic system, I find this part very strange. He gives the example of a hard rubber ball being rolled in a triangle on a flat surface, the point is that if you take the ball out through the triangle and back to...

Homework Statement 4. Implement a simple method to find the square root of a double precision floating point number x. A simple method is to consider the error produced by a “guess” y of the solution. Square the value y and compare with the value x. If y is correct, the error e=|y2-x| where ||...

Homework Statement Here's a screenshot of the problem: http://puu.sh/2Bta5 Homework Equations The Attempt at a Solution As can be seen by the screenshot, the answer's already given, but I'm not sure how to go about getting it. This one has me stumped since e-4x and sin(5x) are...

Homework Statement a. Compute the second order Taylor polynomial centered at 2, P2(x), for the function ln(x). b. Estimate the maximum error of the answer to part a for x in the interval [1,2].Homework EquationsThe Attempt at a Solution For part a, I'm thinking that when it says "second...

1. a. Find Taylor series generated by ex2 centered at 0. b. Express ∫ex2dx as a Taylor series. 2. For part a, I just put the value of "x2" in place of x in the general form for the e^x Taylor series: ex: 1 + x + x2/2! + x3/3! + ... ex2: 1 + x2 + x4/2! + x6/3! + ... For part b...

Homework Statement Find the Taylor series of f(x) = x2ln(1+2x2) centered at c = 0. Homework Equations Taylor Series of f(x) = ln(1+x) is Ʃ from n=1 to ∞ of (-1)n-1xn/n The Attempt at a Solution I have worked the problem to (-1)n4nx2n/n I am not sure where to go from here...

Homework Statement Find the Taylor Series for f(x) = 1/(1-6) centered at c=6 Homework Equations ∞ Ʃ Fn(a)(x-a)/n! n=0 The Attempt at a Solution I believe that the nth derivative of 1/(1-6x) is (-6)n-1n!/(1-6x)n+1 So i figured that the taylor series at c=6 would be...

Homework Statement f(x)=\frac{4x}{(4+x^{2})^{2}}Homework Equations \frac{1}{1-x} = \sum x^{n} The Attempt at a Solution How am I supposed to use that equation to solve the main problem. I have the solution but I don't understand how to do any of it. My professor is horrible, been on...

Could someone please explain how does this taylor expansion work: 1/|r-r'| ≈ 1/r+(r.r')/r3 possibly you have to taylor expand twice to get this result, an attempt at which led me nowhere, surely it cannot be this complicated. any useful comment about this would be greatly appreciated...

Homework Statement Let g(x) = \frac{x}{e^x - 1} = \sum_{n=0}^{\infty} \frac{B_n}{n!} x^n be the taylor series for g about 0. Show B_0 = 1 and \sum_{k=0}^{n} \binom{n+1}{k} B_k = 0 .Homework Equations The Attempt at a Solution g(x) = \sum_{n=0}^{\infty} \frac{g^{(n)}(0)}{n!} x^n , but...

Homework Statement A problem from advanced calculus by Taylor : http://gyazo.com/5d52ea79420c8998a668fab0010857cf Homework Equations ##sin(x) = \sum_{n=0}^{∞} (-1)^n \frac{x^{2n+1}}{(2n+1)!}## ##sin(3x) = \sum_{n=0}^{∞} (-1)^n \frac{3^{2n+1}x^{2n+1}}{(2n+1)!}## The Attempt at a Solution...

A paper I'm reading states the that: for positive hermitian matrices A and B, the Taylor expansion of \log(A+tB) at t=0 is \log(A+tB)=\log(A) + t\int_0^\infty \frac{1}{B+zI}A \frac{1}{B+zI} dz + \mathcal{O}(t^2). However, there is no source or proof given, and I cannot seem to find a...

Homework Statement Calculate the Taylor series expansion about x=0 as far as the term in ##x^2## for the function : ##f(x) = \frac{x-sinx}{e^{-x} - 1 + ln(x+1)}## when ##x≠0## ##f(x) = 1## when ##x=0## Homework Equations Some common Taylor expansions. The Attempt at a Solution...

over a Maclaurin series? Also, how do I calculate e^0 using Maclaurin series? I'm getting 0^0.

I am reading about spontaneous symmtry breaking for superconductors and came a cross to this simple statement: Here is the potential for complex scalar field: V = 1/2 \lambda^2 (|\phi|^2 -\eta^2)^2 . Scalar field is small and we can expand its modulus around \eta : \phi(x) =...

About a week ago, I learned about linear approximation from a great youtube video, it was by Adrian Banner and the series of his lectures I think were from his book Calculus LifeSaver. I truly thought it was so beautiful and powerful a concept. Shortly I also got to know the Taylor Series and...

Homework Statement Consider the followign function f(x) = x^-5 a=1 n=2 0.8 \leq x \leq 1.2 a) Approximate f with a tayloy polynomial of nth degree at the number a = 1 b) use taylor's inequality to estimate the accuracy of approximation f(x) ≈ T_{n}(x) when x lies in the interval...

Homework Statement The energy density of electromagnetic radiation at wavelength λ from a black body at temperature T (degrees Kelvin) is given by Planck's law of black body radiation: f(λ) = \frac{8πhc}{λ^{5}(e^{hc/λkT} - 1)} where h is Planck's constant, c is the speed of light, and...

Homework Statement Find the Taylor polynomial approximation about the point ε = 1/2 for the following function: (x^1/2)(e^-x)The Attempt at a Solution I'm trying to get a taylor polynomial up to the second derivate i.e.: P2(×) = (×^1/2)(e^-x) + (x-ε) * [(e^-x)(1-2×)/2(×^1/2)] +...

I am trying to establish why, I'm assuming one uses taylor series, \frac{\partial u}{\partial t}(t+k/2, x)= (u(t+k,x)-u(t,x))/k + O(k^2) I have tried every possible combination of adding/subtracting taylor series, but either I can not get it exactly or my O(k^2) term doesn't work out (it's...

Prove that if $p^T▽f(x_k)<0$, then $f(x_k+εp)<f(x_k)$ for $ε>0$ sufficiently small. I think we can expand $f(x_k+εp)$ in a Taylor series about the point $x_k$ and look at $f(x_k+εp)-f(x_k)$, but what's then? (Taylor series: $f(x_0+p)=f(x_0)+p^T▽f(x_0)+(1/2)p^T▽^2f(x_0)p+...$ => here...

I'm trying to understand how the algebraic properties of the Dirac delta function might be passed onto the argument of the delta function. One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with...

Homework Statement \mu = \frac{mM}{m+M} a. Show that \mu = m b. Express \mu as m times a series in \frac{m}{M} Homework Equations \mu = \frac{mM}{m+M} The Attempt at a Solution I am having trouble seeing how to turn this into a series. How can I look at the given function...

Hello all, I am a senior physics undergraduate student. I have wondered about the Taylor Expansion for a few years now and just have never bothered to ask. But I will now: I know the Taylor Expansion goes like: f(a) + \frac{f'(a)}{1!}*(x-a) + \frac{f''(a)}{2!}*(x-a)^{2} +...

Homework Statement obtain the number r = √15 -3 as an approximation to the nonzero root of the equation x^2 = sinx by using the cubic Taylor polynomial approximation to sinxHomework Equations cubic taylor polynomial of sinx = x- x^3/3!The Attempt at a Solution Sinx = x-x^3/3! + E(x) x^2 =...

the error of a taylor series of order(I think that's the right word) n is given by \frac{f^{n+1} (s)}{n!} (x-a)^n I think this is right. The error in a linear approximation would simply be \frac{f''(s)}{2} (x-a)^2 My question is what is s and how do I find it. Use linear...

Homework Statement The problem has six charges that are at the corners of a regular hexagon in the xy plane, each charge a distance a from the origin. I have already solved for the electric fields in the x and y direction and now am trying to apply an approximation for the field on the x-axis...

Author: John Taylor Title: Classical Mechanics Amazon Link: https://www.amazon.com/dp/189138922X/?tag=pfamazon01-20 Prerequisities: A Lower-Division mechanics course Contents: Upper-Division of undergrad

Author: Edwin F. Taylor (Author), John Archibald Wheeler (Author) Title: Spacetime Physics Amazon Link: https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20 Prerequisities: Contents:

Question: http://i.imgur.com/GsjeL.png Here is my attempt so far: http://i.imgur.com/AyOCm.png Note: I've used m where the question has used j. My attempt is based off some bad notes I took in class so the way I am trying to solve the problem may not be the best. I'm struggling to...

Hi everyone, Is there a certain technique or a program for converting Taylor expansion to summation notation form and vice versa. Thank you in advance.

The Taylor Series of f(x) = exp(-x^2) at x = 0 is 1-x^2... Why is this? The formula for Taylor Series is f(x) = f(0) + (x/1!)(f'(0)) + (x^2/2!)(f''(0)) + ... and f'(x) = -2x(exp(-x^2)) therefore f'(0) = 0? Can someone please explain why it is 1-x^2?

What does it mean to calculate the Taylor series ABOUT a particular point? I understand the formula for the Taylor series but how do you solve it about a particular point for a function? It's the about the particular point that confuses me. Could someone please explain this and provide...

Hello! I was trying to look for a possible expansion of the ln function. The problem is, that there is no expansion that can be used in all points (like there is for e, sine, cosine, etc..) Why do you think that is? To clarify: Let's say i do the MacLaurin expansion of ln(x+1)...

Why in Taylor series we have some factoriel ##!## factor. f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+... Why we have that ##\frac{1}{n!}## factor?

I'm trying to understand how to use Taylor series expansion as a method to solve complex integrals. I would appreciate someone looking over my thoughts on this. I don't know if they are right or wrong or how they could be improved. I suppose that my issue is that I don't feel confident in my...

Homework Statement compute the following limit: ## \displaystyle{\lim_{x\to +\infty} x \left((1+\frac{1}{x})^{x} - e \right)} ## The Attempt at a Solution i wanted to use the taylor expansion, but didn't know what ##x_0## would be correct, as the x goes to ## \infty##. also, i tried to...

Is there a way to find the cosine of i, the imaginary unit, by computing the following infinite sum? cos(i)=\sum_{n=0}^\infty \frac{(-1)^ni^{2n}}{(2n)!} Since the value of ##i^{2n}## alternates between -1 and 1 for every ##n\in\mathbb{N}##, it can be rewritten as ##(-1)^n##...

Homework Statement Last exam in my school this exircise was given: From norweagen: " Decide the Taylor polynomial of second degree of x=0 of the function: f(x) = 3x^3 + 2x^2 + x + 1 I found the Taylor polynomial of second degree to be: 2X^2+X+1, which is correct. If I get an...

Often you use taylor series to approximate differential equations for easier solving. An example is the small angle approximation to the pendulum. My question is: Is there mathematical tool for calculating the error you make as time goes with such an approximation? Because I could Imagine it...

Homework Statement My Calc II final is tomorrow, and although we never learned it, it's on the review. So I have a few examples. Some I can figure out, some I cant. Examples: f(x)=sinh(x), f(x)=ln(x+1) with x0=0, f(x)=sin(x) with x0=0, f(x)=1/(x-1) with x0=4 The only one of those that I was...

Quick question about Taylor expansions that I am getting pretty confused about. Let's say using biot savart I want to find the field of a INFINITE helix (http://en.wikipedia.org/wiki/Helix) along the axis but with very slight displacements of x and y (x+ε,y+ε). Here is a series of steps I will...

Hey, guys. Having problems with this question because I don't exactly know how to begin it. Homework Statement The problem states to: "Find the Taylor polynomial of smallest degree of an appropriate function about a suitable point to approximate the given number to within the indicated...

I recently thought to myself about how a slight modification to the taylor series of e^x, which is, of course: \sum_{n=0}^\infty \frac{x^n}{n!} would change the equation. How would changing this to: \sum_{n=0}^\infty \frac{x^{n/2}}{\Gamma(n/2+1)} change the equation? Would it still be...

let nε Z. the polylogarithm functions are a family of functions, one for each n. they are defined by the following taylor series: Lin(x)= Ʃ xk/kn 1.calculate the radius of convergence [b]3. when i attempted this part, i couldn't use theratio or root test, so by comparison i got R=∞...

Hello Everyone! Suppose $f(x)$ can be written as $f(x)=P_n(x)+R_n(x)$ where the first term on the RHS is the Taylor polynomial and the second term is the remainder. If the sum $\sum _{n=0} ^{\infty} = c_n x^n$ converges for $|x|<R$, does this mean I can freely write $f(x)=\sum _{n=0} ^{\infty}...

In many of my physics classes we have been using Taylor Expansions, and sometimes I get a bit confused. For example, I feel like different things are going on when one expands (1-x)^-2 vs. e^(-Ax^2), where I just have some constant in front of x^2 to help make my point. To keep things simple...