Taylor Definition and 878 Threads

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+...

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 Let f(x)=x2 +3x -5, and let the summation (from n=0 to infinity) an (x-4)n be the Taylor series of f about 4. Find the values of a0, a1, a2, a3, and a4. Homework Equations The Attempt at a Solution What am I supposed to do with the summation? And what does it mean...

Homework Statement use the third degree Taylor polynomial of cos at 0 to show that the solutions of x2=cos x are approx. \pm\sqrt{2/3}, and find bounds on the error. Homework Equations P2n,0(x) = 1-x2/2!+x4/4!+...+(-1)nx2n/(2n)! The Attempt at a Solution when it says "third...

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...

Homework Statement Find the Taylor Polynomial of degree 3 for the function f(x) = ln3x about a = 1/3Homework Equations NoneThe Attempt at a Solution I have found up to the fourth derivative of f(x) along with the values of the derivatives at x = 1/3. At this point i get Σ{(-1)kk!fk(1/3)}, but...

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...

Homework Statement It seems that I'm a little bit lost about this exercise. It says: Find the taylors polynomial of third degree centered at the origin for z=\cos y \sin x. Estimate the error for: \Delta x=-0.15,\Delta y=0.2. So, I did the first part (the easy one), the taylors polynomial for...

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...

Homework Statement I'm reading about fluid mechanics and in one of the examples they have approximated the velocity field. The field is two dimensional u = (u,v) I have never seen this before so cold someone tell me what it is called so I can look it up? The notes I am reading are hand...

Homework Statement If fd is the fractional uncertainty on the distance, what is the fractional uncertainty on the total mass fM? Hint: use our Taylor expansion approximation that (1 +- x) ^a ~ 1 +- ax when x << 1. The fractional uncertainty in the mass fM will be related to fd in a simple...

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...

So this was a textbook problem my professor did in lecture. I felt like I followed along with the logic as she went along, but after a few days and looking back it, I can't seem to recreate it genuinely. Homework Statement A ball is thrown with initial speed v0 up an inclined plane. The...

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...

hello, please help to calculate the taylor polynomial for http://latex.codecogs.com/gif.latex?f(x)=x^{x}-1 around the point a=1 i thought to write it as g(x)=x^x and then f(x)=g(x)-1 and then find the polynomial for g(x) as lng(x)=xln(x) but it seems incorrect.

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...

I am wroking through an electrodynamics textbook and there is this Taylor expansion to do later a multipole expansion. But I can't figure out how the author does it. Please any help? the expansion: \frac{1}{|\vec{r}-\vec{r'}|} = \frac{1}{r} - \sum^3_{i=1} x'_i \frac{\partial}{\partial...

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?

«Common sense is the collection of prejudices acquired by the age of 18» Albert Einstein. At the very beginning of my small article I would like to pay attention of the reader to R. Penrose’s rather new book «Big, small and human reason» publishing house the Amphora, St.-Petersburg 2008...

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...

I have a couple of questions about exercise **103 (yes, a two-star problem!) in Taylor & Wheeler's Spacetime Physics. In part (a), it says "For an atom \beta_r \leq Z/137 (Ex. 101), and for small Z, \beta_r \ll 1. Therefore \tan(d\phi) \approx d\phi \approx -\beta_r^2 \sin(\alpha)." But it's...

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...

I have the following problem: Assume g is a (smooth enough) function, X a random variable and \varepsilon^h a sequence of random variables, whose moments converge to 0 as h goes to zero. I would then like to prove that \mathbb{E}\left|g(X+\varepsilon^h)-g(X)\right| converges to zero as well...

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...