Taylor Definition and 849 Threads

Find the function that has the following Taylor series representation: \sum^{\infty}_________{m=0}\frac{(m+s)^{-1}x^{m}}{m!} Where s is a constant such that 0<Re(s)<1. Any ideas?

Homework Statement 1. Use Taylor's Theorem to determine the accuracy of the approximation. arcsin(0.4) = 0.4 + \frac{(0.4)^{3}}{2*3}} 2. Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value fo x to be less...

I'm reading a paper on tissue cell rheology ("Viscoelasticity of the human red blood cell") that models the creep compliance of the cell (in the s-domain) as J(s) = \frac{1}{As+Bs^{a+1}} where 0\leq a\leq 1. Since there's no closed-form inverse Laplace transform for this expression, they...

When approximating a function with a Taylor series, I understand a series is centered around a given point a, and converges within a certain radius R. Say for a series with center a the interval of convergence is [a-R, a+R]. Does this imply that: 1. There also exists a Taylor series expansion...

Hello, I am looking for a resource (preferably a textbook) to help me with nonlinear, multivariable functions and working through taylor series expansions of them. My calculus book only covers single variable expansions unfortunately. Thanks

Question about Taylor series and "big Oh" notation Can someone please explain WHY it's true that e^x = 1 + x + \frac{x^2}{2} + \mathcal{O}(x^3) I'm somewhat familiar with "big Oh" notation and what it stands for, but I'm not quite sure why the above statement is true (or statements...

Homework Statement (a) Give Taylor Polynomal of order 4 for ln(1+x) about 0. (b) Write down Tn(x) of order n by looking at patterns in derivatives in part (a), where n is a positive integer. (c) Write down the remainder term for the poly. in (b) (d) How large must n be to ensure Tn gives a...

Homework Statement f(x) = \frac{1-cos(X^2)}{x^3} which identity shoud i use? and tips on this type of questions? once i can separate them, then i'll be good thanks!

you know this, right? f(x) = \sum^{\infty}_{k=0} \frac{f^{(k)}(x_0) (x-x_0)^k}{k!} for an analytic function, at x0 = 0, you have to say that 0^0 equals 1 for the constant term. if 0^0 is indeterminate then how can you just say it's 1 in this case?

Homework Statement I want to know that how to calculate the required number of terms to obtain a given decimal accuracy in two variable Taylor series . In one variable case i know there is an error term R(n)=[ f(e)^(n+1)* (x-c)^(n+1)] / (n+1)! where 'e' is...

[b]1. Hi, I am new to taylor series expansions and just wondered if somebody could demonstrate how to do the following. Find the Taylor series of the following functions by using the standard Taylor series also find the Radius of convergence in each case. 1.log(x) about x=2...

Homework Statement (a) Use Taylor's theorem with the Lagrange remainder to show that log(1+x) = \sum^{\infty}_{k=1}\frac{(-1)^{k+1}}{k}x^{k} for 0<x<1. (b) Now apply Taylor's theorem to log(1-x) to show that the above result holds for -1<x<0. Homework Equations Taylor's...

In this: http://www.math.tamu.edu/~fulling/coalweb/sinsubst.pdf It says that to find the Taylor series of sin(2x + 1) around the point x = 0, we cannot just substitute 2x+1 into the Maclaurin series for sinx because 2x + 1 doesn't approach a limit of 0 as x approaches 0. It says we have...

1. The problem \statement, all variables and given/known data Estimate the error involved in using the first n terms for the function F(x) = \int_0^x e^{-t^2} dt Homework Equations The Attempt at a Solution I am using the Lagrange form of the remainder. I need to know the n+1 derivative of...

[b]1. Use Taylor's expansion about zero to find approximations as follows. You need not compute explicitly the finite sums. (a) sin(1) to within 10^-12; (b) e to within 10^-18: [b]3. I know that the taylor expansion for e is e=\sum_{n=1}^{\infty}\frac{1}x^{n}/n! and I aslo know that...

Homework Statement A water wave has length L moves with velocity V across body of water with depth d, then v^2=gL/2pi•tanh(2pi•d/L) A) if water is deep, show that v^2~(gL/2pi)^1/2 B) if shallow use maclairin series for tanh to show v~(gd)^1/2 Homework Equations Up above [b]3. The...

Find P5(x), the 5th order Taylor series, of sin (x) about x = 0. Hence find the 4th order Taylor series for x sin (2x) about x = 0. In this question why is it required to find the 5th order taylor series of sin(x) to find the 4th order taylor series of xsin(2x)?

Sorry, the title should be: geometric intepretation of moments My question is: does the formula of the moments have a geometrical interpreation? It is defined as: m(p) = \int{x^{p}f(x)dx} If you can't see the formula it is here too: http://en.wikipedia.org/wiki/Moment_(mathematics) with c=0...

Homework Statement Could someone please explain how the taylor expansion of 1/(r-r') turns into ( 1/r+(r'.r)/r^3 + (3(r.r')^2-r^2r'^2)/2r^5 +...) Homework Equations The Attempt at a Solution

How do you Taylor expand e^{i \vec{k} \cdot \vec{r}} the general formula is \phi(\vec{r}+\vec{a})=\sum_{n=0}^{\infty} \frac{1}{n!} (\vec{a} \cdot \nabla)^n \phi(\vec{a}) but \vec{k} \cdot \vec{r} isn't of the form \vec{r}+\vec{a} is it?

Homework Statement I'm having a hard time following a taylor expansion that contains vectors... http://img9.imageshack.us/img9/9656/blahz.png http://g.imageshack.us/img9/blahz.png/1/ Homework Equations The Attempt at a Solution Here's how I would expand it: -GMR/R^3 -...

Homework Statement Prove if t > 1 then log(t) - \int^{t+1}_{t}log(x) dx differs from -\frac{t}{2} by less than \frac{t^2}{6} Homework Equations Hint: Work out the integral using Taylor series for log(1+x) at the point 0 The Attempt at a Solution Using substitution I get...

Homework Statement Let f be a function with derivatives of all orders and for which f(2)=7. When n is odd, the nth derivative of f at x=2 is 0. When n is even and n=>2, the nth derivative of f at x=2 is given by f(n) (2)= (n-1)!/3n a. Write the sixth-degree Taylor polynomial for f about...

Homework Statement The Taylor polynomial of degree 100 for the function f about x=3 is given by p(x)= (x-3)^2 - (x-3)^4/2! +... + (-1)^n+1 [(x-3)^n2]/n! +... - (x-3)^100/50! What is the value of f^30 (3)? D) 1/15! or E)30!/15! Homework Equations The Attempt at a...

an idea i had: factorizing taylor polynomials Can any taylor polynomial be factorized into an infinite product representation? I think so. I was able to do this(kinda) with sin(x), i did it this way. because sin(0)=0, there must be an x in the factorization. because every x of...

Homework Statement Q1) Use the Taylor series of f (x), centered at x0 to show that F1 =[ f (x + h) - f (x)]/h F2 =[ f (x) - f (x - h) ]/h F3 =[ f (x + h) - f (x - h) ]/2h F4 =[ f (x - 2h) - 8 f (x - h) + 8 f (x + h) - f (x + 2h) ]/12h are all estimates of f '(x). What is the error...

Can someone please tell me how to expand \ln(x + \sqrt{1+x^2}) for small x? I'd like to retain terms at least up to order x^5. Thanks!

Homework Statement Determine the Taylor Series for f(x) = ln(1-3x) about x = 0Homework Equations ln(1+x) = \sum\fract(-1)^n^+^1 x^n /{n}The Attempt at a Solution ln(1-3x) = ln(1+(-3x)) ln(1+(-3x)) = \sum\fract(-1)^n^+^2 x^3^n /{n} Is that right?

Hi. How can I derive the Taylor series expansion and the radius of convergence for hyperbolic tangent tanh(x) around the point x=0. I can find the expression for the above in various sites, but the proof is'nt discussed. I guess the above question reduces to how can I get the expression...

I know that the Taylor Series of f(x)= \frac{1}{1+x^2} around x0 = 0 is 1 - x^2 + x^4 + ... + (-1)^n x^{2n} + ... for |x|<1 But what I want is to construct the Taylor Series of f(x)=...

Okay so suppose I have the Initial Value Problem: \left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{IVP} NB. I am considering only real functions of real variables. If f(x,y) is...

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:)

Homework Statement Develop the Taylor expansion of ln(1+z). Homework Equations Taylor Expansion: f(z) = sum (n=0 to infinity) (z-z0)n{f(n)(z0)}/{n!} Cauchy Integral Formula: f(z) = (1/(2*pi*i)) <<Closed Integral>> {dz' f(z')} / {z'-z} The Attempt at a Solution I have NO idea...

prove this inequality for x>0 x-\frac{x^3}{6}+\frac{x^5}{120}>\sin x this is a tailor series for sin x sinx=x-\frac{x^3}{6}+\frac{x^5}{120}+R_5 for this innequality to be correct the remainder must be negative but i can't prove it because there are values for c when the -sin c...

My professor just told me that if \Delta x is small, then we can expand L(x+\Delta x) as follows: L(x + \Delta x) = L(x) + \frac{d L}{d x} \Delta x + \frac{1}{2!} \frac{d^2 L}{d x^2} (\Delta x)^2 + \ldots, where each of the derivatives above is evaluated at x. Could someone please...

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??

Homework Statement Hey guys. I need to develop Taylor series for this function (cos(z) * cosh(z)). I know the Taylor development for cos and the Taylor development for cosh but I have no idea how to combine the two, if it's possible, any idea guys? And another thing, does it matters if we...

Homework Statement Basically, I have a differential equation. One of the elements of it is... F(P) = 0.2P(1 - (P/10)) And I need to replace it with it's first-order Taylor polynomial centered at P=10. The Attempt at a Solution I haven't done Taylor polynomial stuff in over a...

Can anyone provide me with a website that has copies of the original works of Riemann, Taylor, famous mathematicians. I am looking for papers on proved theorems.

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

Homework Statement With n>1, show that (a) \frac{1}{n}-ln\frac{n}{n-1}<0 and (b) \frac{1}{n}-ln\frac{n+1}{n}>0 Use these inequalities to show that the Euler-Mascheron constant (eq. 5.28 - page330) is finite. Homework Equations This is in the chapter on infinite series, in the section...

Hi Guys, I was wondering if it is possible (why or why not) to define the floor function, Floor[x], as an infinite Taylor Series centered around x=a? Any sort of help is greatly appreciated! flouran

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.

Hey all, so I need to find 4th degree taylor polynomial of f(x)=sec(x) centered at c=0 Can I just use substitution to find the answer since sec(x) = 1/cos(x) and I know the taylor series for cos(x). I guess, essentially, can I take the reciprocal of the taylor series of cosx to get sec(x)...

Hey all, so I need to find 4th degree taylor polynomial of f(x)=sec(x) centered at c=0 Can I just use substitution to find the answer since sec(x) = 1/cos(x) and I know the taylor series for cos(x). I guess, essentially, can I take the reciprocal of the taylor series of cosx to get sec(x)...

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...