Taylor Definition and 849 Threads

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...

Homework Statement Hi there. I have this exercise which I'm trying to solve now. It says: Using that \displaystyle\sum_{n=0}^{\infty}x^n=(1-x)^{-1} find one Taylor development for the function f(x)=\ln(1-x) So, I've made some derivatives...

Solved Thanks guys

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 I have to approximate sin(1/2) with the taylor inequality Homework Equations taylors inequality |Rn(x)| ≤ M/(n+1)! | x-a|n+1 The Attempt at a Solution Im not really sure what the significance of this is, but ill do the derivatives f(x) = sin(x) f'(x) = cos(x) f''(x) =...

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...

Homework Statement Determine the limit and then prove your claim. limx\rightarrow\infty (1+\frac{1}{x^2} }) xHomework Equations I know that the formal definition that I need to use to prove the limit is: {limx\rightarrow\infty (1+\frac{1}{x^2})x=1}={\forall \epsilon>0, \exists N > 0, \ni x>N...

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 Let f be a function that has derivatives of all orders for all real numbers. Assume f(1) = 3, f'(1) = -2, f''(1) = 2, and f'''(1) = 4 a. Write the second-degree Taylor polynomial for f about x = 1 and use it to approximate f(0.7) b. Write the third-degree Taylor...

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 In my never ending quest to suck and never be able to do Taylor Expansions, I have another one. I hope one day I'll be able to do these. I have an unknown material and a scanning tunnel microscope. A layer of hydrogen atoms of radius R are added to the surface. This of...

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 I have this equation: T=(1+\frac{U_{0}^{2}}{4E(U_{0}-E)}sinh^{2}(2 \alpha L))^{-1} Where α is given by: \alpha = \sqrt{ \frac{2m(U_{0}-E)}{\hbar^{2}}} I have to show that in the limit αL>>1 my equation is approximately given by...

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) =...

One of the greatest movie legends we've ever had. http://news.yahoo.com/s/ap/us_obit_taylor

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...

I'm a 2nd year Physics undergrad and am ashamed to admit I don't fully understand the taylor expansion. I have seen the Maclaurin expansion derived which I understand but the Taylor expansion is a little weird to me. I don't really understand what it means to expand a function "about a...

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?

Homework Statement Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder. The Attempt at a Solution I found the polynomial to be 1 + x + x2 + x3 + x4 + x5, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2]...

Well - I am a 16 year old student, whos really interested in math. I do a lot of studying on my own, because I am a bit bored with the present math in school. Right now I am reading about solving differential equations with power series. I can do this, and i do understand the recurrence...

Homework Statement find the 2nd, 3rd, and 6th degree taylor approximation of: f(x) = 10(x/2 -0.25)5 + (x-0.5)3 + 9(x-0.75)2-8(x-0.25)-1 for h = 0.1 to h = 1, with \Deltah = 0.05 and where xo=0; and x = h Homework Equations N.A The Attempt at a Solution I just need to...

I want to show this taylor expansion: \frac{1}{\sqrt{1+{x}^{2}}} \rightarrow x^2 what I keep getting is something to the x^3 could some one please help me with this simple expansion?

Homework Statement Find the Taylor polynomial of degree 3 of \frac{1}{2+x-2y} near (2,1). Homework Equations The Attempt at a Solution I have already solved this problem by evaluating the R^2 Taylor series; I'm mostly curious about another aspect of the problem. By substituting u = x-2y, it...

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...

I'm aiming to calculate ln(x) numerically. I'm using the following procedure for this: 1) If x is greater or equal to 1, use Newton's method. 2) If x is smaller between 0 and 1, use Taylor series expansion. Newton's method works good, but I have problems with Taylor series expansion method...

Had a recent homework questions: Find a bound for the error |f(x)-P3(x)| in using P3(x) to approximated f(x) on the interval [-1/2,1/2] where f(x)=ln(1+x) abd P3(x) refers to the third-order Taylor polynomial. I found the Taylor series of f(x) seen below: x- x^2/2!+(2x^3)/3! I know...

ok .lets say the expression we have is ex the taylor expansion becomes 1+x+x2/2+... integrating becomes x+x2/2+x3/6+...+c so how do we know that c = 1? for it to become back to ex becos it is said that integral of ex = ex do we just let x be 0 to find c = 1? does it work for all...

So we can use the Taylor's theorem to come up with a Taylor series represent certain functions. This series is a power series. So far (I'm in my second year of calc, senior in high school), I've never seen a power series that wasn't a Taylor series. So are all power series taylor series? Whether...

Hello, I'm taking my first calculus course right now, and something struck me regarding the remainder in integral form of a Taylor series expansion: Let's say we have a Taylor expansion of the (n-1):th order, which has a remainder of the form Now, my claim is that if we integrate by...

Homework Statement With a Taylor series expansion of the well-behaved \rho ({\bf{x'}}) around {\bf{x'}} = {\bf{x}}, one finds the Taylor expansion of the charge density to be, \rho ({\bf{x'}}) = \rho ({\bf{x}}) + {\textstyle{1 \over 6}}{r^2}{\nabla ^2}\rho + ... Homework Equations...

Homework Statement How do we get that the Taylor Series of 1/(1+x^2) around x= 0 is 1 - x^2 + x^4 + ... + (-1)^n x^{2n} + ... for |x|<1, without using a substitution of x=-x^2 into the Taylor series for 1/(1-x)? Homework Equations The Attempt at a Solution

Hello. For a physics course, I need to often make use of the binomial series and it's corollary, the expansion of: \sqrt{1-x^2} This probably sounds rather stupid, but for some reason, when I do a MacClaurin expansion of this series, I cannot seem to generate the correct series, which I...

I have a homework question like this. "Find the taylor series of the function f(x) = (x2+2x+1)/(x-6)2(x+2) at x=2" I'm trying to simplify this expression so I can take the derivative. I only got this far: (x+1)(x+1)/(x-6)(x-6)(x+2) Can this be simplified more so that I can easily...

Homework Statement I have the function (a(1+z)3 + b)-1/2 and i need to taylor expand it around z=0 to the first order, a and b are constants, there sum is equal to one. I have the answer: 1 - (1+q)z where q = a/2 - b This is in my physics book but it does not explain the...

So I'm studying for a final, and it just so happens my professor threw taylor polynomials at us in the last week.. I understand the concept of a taylor polynomial but i need some help fully understand the LaGrange remainder theorem if we have a function that has n derivatives on the interval...

Homework Statement [sorry about the formatting, I had no idea how I would latex the sigma notation] Let f(x) = [n=1 to infinity] summation of (-1)n n2 / 3n * (x+1)n Find the Taylor series of f(x) centered at c = -1Homework Equations Taylor series defined by [n=0 to infinity] summation of...

Homework Statement http://img704.imageshack.us/f/helpppp.png/ Homework Equations The Attempt at a Solution I know e^(x) = 1 + x + x^(2)/2! + ... But if you multiply that by (x^(4))+4x^(3)) How do you know what bn and a is?

Hi, Im really stuck on my homework . The question is : For what values of x do you expect the following Taylor series to converge? Do not work out the series . (a) sqrtX^2-x-2 about x=1/3 b) sin(1-x^2) about x=0 for a) I've put no vlues of x would the series converge. is this correct? and...

Hi everybody, I hope anyone could help Homework Statement Find the first three terms of the Taylor series for f(x) at c. http://dc12.arabsh.com/i/02388/kgybq4dwkug3.png Homework Equations f(x)= f(c) + f'(c).(x-c)/1! + f"(c).(x-c)^2/2! + f'''(c).(x-c)^3/3! +...+...

Hello, I need help with this problem. I need to find the first three terms of the Taylor series for the function f(x)= (1 + x)^(1/3) to get an estimate for 1.06^(1/3). Hence I did: f(x)= (1 + x)^(1/3) f'(x)= (1/3)(1 + x)^(-2/3) f''(x)= (-2/9)(1 + x)^(-5/3) f(a) + f'(x)/1! * (x - a) +...

Let f(x) = \frac{4-4x}{4x^{2} -8x -5}; given the partial decomposition, \frac{4-4x}{4x^{2} -8x -5} = \frac{1}{5-2x} - \frac{1}{1+2x}, find the Taylor series of f(x) about 1. Express your answer in sigma notation and simplify as much as possible. Dtermine the open interval of...

Hello, I am trying to come up with an expression for a bound on the sum of higher order terms, above second order. Consider the following Taylor expansion of a function f(x) around a point a, f(x) = f(a) + \frac{f^{(1)}(a)}{1!}(x-a) + \frac{f^{(2)}(a)}{2!}(x-a)^2+...