Taylor series Definition and 492 Threads

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...

Homework Statement The equation for the velocity of a falling parachutist can be computed by, v(t) = \frac{gm}{c}(1-e^{-(\frac{c}{m})t}) Use a first-order error analysis to estimate the error of v at t = 6, if g = 9.8 and m = 9.8, c = 12.5 plus or minus 1.5. Homework Equations...

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...

Homework Statement See figure attached, Homework Equations The Attempt at a Solution Isn't the Maclaurin series just simply the Taylor series around 0? (\text{i.e. } (x-c), \quad c=0, \quad x ) Also for part B, how do we go about solving for | \epsilon_{t} |? Thanks...

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...

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 [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+...

Homework Statement Let f(x) = x3ln(1+x2), and let the summation (from n=0 to infinity) anxn be the Taylor series for f about 0. Then what is a3? Homework Equations The Attempt at a Solution What?! I definitely don't expect the answer, but does anyone know how I could go about...

Homework Statement find Taylor polynomial for ln x of degree n, at 2 (Pn,2(x)) Homework Equations Pn,1(x)= (x-1) - (x-1)2/2 + ... + (-1)n-1(x-1)n/n The Attempt at a Solution there doesn't seem to be an obvious pattern to this. the coefficients for n=1 to n=4 are 2, -8, 24, -64...

I have a shaky understanding of problems concerning Taylor Series. For example, the question below. Let f(x)=\tan^{-1}\left(\frac{1+x}{1-x}\right) where -\frac{1}{2}\leq x \leq \frac{1}{2}. Find the value of f^{2005}(0) the Taylor Series of \tan^{-1} is...

The formula given by my instructor for a Taylor Series approximation of the second order at point (a,b) is f(a,b) + grad(f(a,b))x + 1/2 H(f(a,b)) x If you recognize this formula, do you know what the x vector is? Note: x is the x-vector, and H represents the Hessian Matrix. Thanks! The...

Hi! I am a 16 year old trying to figure out the application of taylor series. I understand most of its uses when applied to functions like e^x, sinx, cosx, but in a mechanics book, i am required to find delta-F, a finite change in a function F. Ostensibly, this appears to be a step that needs...

When it says "about a point x=a", what does this mean? why not just say at x = a? Thanks

Homework Statement Obtain the Taylor series in powers of x + 1 for f(x) = x/(2 + x), giving the general term. Homework Equations The Attempt at a Solution Wrote it out as x*(1/1-(-(x+1)).

Homework Statement This is actually not a problem, it's something in my notes. The function I am supposed to be approximating is V(x) = V0(1 - ex/a)2 - V0 V0 and a are constants. Homework Equations The Attempt at a Solution It says that the function given is not a parabola. But it can be...

Hello, I am trying to evaluate the series \sum{\frac{x^n}{n!}e^{cn^2}} where c is a constant. I think this problem is equivalent to find f(x) such that \frac{d^{n}f(0)}{dx^{n}} = \frac{e^{cn^{2}}}{n!} I believe this must be a modified exponential since for c=0, it reduces to...

Homework Statement See figure attached. Homework Equations The Attempt at a Solution Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the, \frac{1}{x^{2}} \int x^{-2} = \frac{-1}{x} + C How do I...

Hello, if I understand correctly the Taylor approximation for a=0 gives me the possibility to approximate a function, say sin(x), at any x. But, what gives me Taylor polynomial at some point http://latex.codecogs.com/gif.latex?a\neq0 ,[/URL] what's the difference? what does it mean centred...

I am trying to linearize a function, f(x), where x is a normally distributed N(0,1) random variable. How can I perform a taylor series expansion around a deterministic value x0? Thanks.

Homework Statement The velocity of a proton relative to our galaxy is vp/c = 1-(0.5*10^20), i.e. almost one. Such protons are actually observed. When velocity it very nearly one \gamma is very large. 1/\gamma is very small. Use Taylor series to show that for v almost one we have...

Can anyone help me for the leading order terms in the taylor series for the function f(x,y) = Sqrt(a*x^8+b*x^4*y^4+y^8), centered at x=0,y = 0 and a,b,c constants?

where do a multiple Taylor series converge ?? i mean if given a function f(x,y) can i expand this f into a double Taylor series that will converge on a rectangle ? for example , if one can ensure that it converges for |x| <1 and |y| <1

Hello all, I am currently studying multivariable calculus, and I am interested in the Taylor series for two variable function. I am not sure where to begin; I cannot understand any of the proofs (which are apparently sparse) on the internet; they all just state it using a sigma sum; not...

Homework Statement Let f(z)=\sum_{n=0}^{\infty} a_n z^n be analytic at {z: |z|<R} and satisfies: |f(z)| \leq M for every |z|<R. Let's define: d=the distance between the origin and the closest zero of f(z). Prove: d \geq \frac{R|a_0|}{M+|a_0|} . Hope you'll be able to help me...

Hi, We need a generic expression of a taylor series nth term to find out the radius of convergence of the series. However, there are series where I don't think it is even possible to find a generic term. How do we find the radius of convergence in such cases? e.g. sqrt (1 - x^2) There...

"Partial" Taylor Series Expansion It has been awhile since I have had to use a Taylor series expansion (from scratch). I looked it up on wiki and the rules are easy enough, I am just a little confused as to how I apply it to a multivariable function, but only expand it about one variable...

I'm doing some review over summer before starting college, and one of the practice exams has a question pertaining to the remainder of a taylor series Homework Statement Show that \left|\cos{(1+x)}-\{\cos{(1)}(1-\frac{x^2}{2})-\sin{(1)}(x-\frac{x^3}{3!})\}\right|<\frac{1}{15000} for |x|<0.2...

Taylor Series Expansion About the Point "i" Homework Statement Calculate the radius of convergence of the Taylor series for \frac{1}{z^2-2z+2} about the point i. The Attempt at a Solution I can find the radius of convergence if I can determine the expansion but I can't seem to...

i can't understand how the got this variation of taylor series formula f(x+h)=\sum_{k=0}^{\infty}\frac{f^{(k)}(x)}{k!}(h)^k http://mathworld.wolfram.com/TaylorSeries.html when around some point there is no x-x_0

Homework Statement From the taylor series we can replace x =x_{0} + h but how does \delta f = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0}) become \delta f = hf(x_{0}, y_{0}) + kf(x_{0}, y_{0}) I can see the first step, but how do you get it to the second?Homework Equations The Attempt at a Solution

f(x)=sinx taylor series centered at pi/2 sum((-1)^n (x-pi/2)^(2n)/(2n)! , n=0,infty ) with radius of convergence infty

f(x)=1/(1-x^2)^(1/2) 1/x^(1/2)=1+ sum(( (-1)^n 1*3*5*7...(2n-1)(x-1)^n )/(2^n n! ) , n=1, infty ) thus 1/(1-x^2)^(1/2) = 1+ sum(( 1*3*5*7...(2n-1)(x^2)^n )/(2^n n! ) , n=1, infty ) is this a correct taylor series representation centered at 1

I was just pondering today how the kinematic equation for position looks like a taylor expansion. x = x0 + dx/dt *t + (1/2)*d2x/dt2*t2 I believe it continues like that, exactly like a taylor expansion does, so the next term would be (1/6)*d3x/dt3*t3 If it is indeed a taylor expansion, what...

Hello, I was wondering if anyone could explain to me the thought process behind how you find the maximum remainder of a Taylor series? I read the wiki article and didn't help me at all, http://en.wikipedia.org/wiki/Taylor's_theorem My book talks about something like this(image is...

Homework Statement For f(x) = xln(x), find the taylor series expansion of f(x) about x = 1, and write the infinite series in compact form. 2. The attempt at a solution I can find the expansion itself fine, these are the first few terms: 0 + (x-1) + \frac{(x-1)^{2}}{2!} -...

so F = mgR2/(R+h)2 where R is the radius of the earth. consider the situation where h is much smaller than R. a) show that F is approximately equal to mg b)express F as mg multiplied by a series in h/R so i need help on getting started. would showing that F is approximately equal...

Homework Statement find the taylor series for the function f(x) = \frac{x^2+1}{4x+5} Homework Equations N/A The Attempt at a Solution how to do this? 1st attempt. i did turn it this term \frac{x}{4} + \frac{-5x+4}{16x+20} can i turn this to taylor series? maybe i know how to make...

I don't have anyone else to ask. So I have to ask you guys. I learned about Taylor series, and then I went back and looked at linear and quadratic approximations, and they are Taylor series except only taken so far. I'm pretty much just looking for confirmation on my idea, it seems perfect.

Homework Statement find taylor series for \frac{x-1}{1+x} at x=1 Homework Equations The Attempt at a Solution how to change this form \frac{x-1}{1+x} to something like this \frac{1}{1+a} or \frac{1}{1-a} help me please T_T or should i do like this \sum\frac{f^n(1)(x-1)^n}{n!} and find...

Homework Statement Find the Taylor Series for f(w) = 1/w centered at w0 = 1 using 1/w = (1/1 + (w-1)). Show that the series converges when |w-1| < 1 Homework Equations use 1/w = (1/1 + (w-1)) The Attempt at a Solution

Homework Statement This is a three part question: It is based off the first two sections. I'm pretty sure the first two answers are correct, but I have no idea how to do the third question. Write the First three nonzero terms and the general term of the Taylor series expansion about x=0...

Homework Statement If \sum_{n=0}^{\infty} a_{n}x^n is a Taylor series that converges to f(x) for all real x, then f'(1) = ? Homework Equations A Taylor series: \sum_{n=0}^{\infty} \frac {f^{(n)}(c)}{n!}(x-c)^n and the dirv of a Taylor series: f'(x)=\sum_{n=0}^{\infty}...