Taylor Definition and 878 Threads

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?

Homework Statement f(E) = \left(\frac{E_c}{E} \right)^{1/2} + \frac{E}{kT} Expand this as a Taylor function with the form... f \approx a_0 + a_1(E-E_0) + a_2(E-E_0) Hint being a_1 will be 0, because E_0 is a Gamow peak in this case, so slope will be 0. What I need to do is...

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...

Consider the IVP: \left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{ze IVP :p} Hypothesis: f(x,y)\subset C^\infty_{x,y}(D)\; \; / \; \;(x_0,y_0)\in D [Note that this condition automatically satisfies the hypotheses of the...

Questions: Is there a quicker way to find the formula for the nth derivative of a function, instead of finding the first several derivatives and trying to find a pattern, and using that pattern to form the equation for the nth derivative? Also, is there a formula for the nth derivative...

I was wondering if someone could help me with Goldstein's equation 6.3 (3rd Edition). It is the chapter of oscillations and all that he has done in the equation is to expand it in the form of a Taylor series. I can't seem to get how all those ni's come to get there.

Homework Statement Find the taylor series of \frac{1+z}{1-z} where z is a complex number and |z| < 1 Homework Equations \sum^{\infty}_{0} z^n = \frac{1}{1-z} if |z| < 1 The Attempt at a Solution \sum^{\infty}_{0} z^n = \frac{1}{1-z} \frac{1+z}{1-z} =...

Homework Statement 1/(4x-5) - z/(3x-2) based @ 0, answers are in those z things.. sigma Homework Equations i think we use sigma of e^x, but idk how... The Attempt at a Solution since tayor sereis of e^x is like 1/x, do i plug 4x-5 in? thanks

Homework Statement sin(x)= sum((-1)^k* (x^(2k+1)/(2k+1)!))k=0 to infinity Homework Equations so if i want to find sin(x)^2, (not sin(x^2), that would be easier though...) The Attempt at a Solution then... do i square the whole thing, like this? sum(((-1)^k*...

Homework Statement how to you find like the answer for f(1.5), or f(1.00001) those kind of question? thanks with like eq. = f(b)(x-b)... am i making sense? thanks

Homework Statement Need to calculate fractional uncertainty f, of M (mass of a star in this case), where f is much less than one. The hint i was given was all i need to know is M \alpha d3, and use a taylor expansion to the first order in f. M = mass of a star, d = distance to star...

not sure I get the Taylor Series... Hello Everyone. I understand that the taylor series approximate a function locally about a point, within the radius of convergence. If we use the Taylor series it means that we do not know the function itself. But to find the taylor series we need the...

Homework Statement What is the quadratic approximation to the potential function? Homework Equations U(x) = U0((a/x)+(x/a)) U0= 20 a=4 The Attempt at a Solution This is just the last part of a question on my engineering homework, I never learned Taylor expansions before even...

Homework Statement find the first four nonzero terms in the power series expansion of tan(x) about a=0 Homework Equations \Sigma_{n=0}^{\infty} \frac{f^n (a)}{n!}(x-a)^n The Attempt at a Solution Well the series has a zero term at each even n (0,2,4 etc) for n=1 I got x, for...

Homework Statement I can either use the alternating series estimation thereom (which i don't really know) or Taylor's Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. sin(x) = x - (x^3)/6 (|error| < 0.01) Do I...

Homework Statement Find the Taylor polynomial T_n(x) for the function arcsin x at a = 0, n = 3 Homework Equations Well, I understand the Taylor poly. for sine, but how do i get arcsine?

Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct. Please could you also guide me on Q1 & 4. Q1...

Homework Statement using the Taylor Formula, find the series for the function f(x)=e^{2x}Homework Equations \sum \frac{f^{n}(a)}{n!} (x-a)^{n} any help as to where i start would be great. new to series...

Hullo, Somehow, I couldn't get the TeX to come out right. I have been trying to learn scheme theory (algebraic geometry) and completely forgotten how to do this simple calculus type stuff... Homework Statement Let V be a potential of the form [tex]V = \left(\frac{1}{r} +...

Many of you have probably used the book Differential Equations by Lomen & Lovelock. For my class I'm working on Example 2, Page 153. You don't need to see the book, though, to help me out. It's a four-part problem and I'm on the last step not knowing where to take it. In Part B, we...

Homework Statement Let T_(4)(x): be the Taylor polynomial of degree 4 of the function ` f(x) = ln(1+x) ` at `a = 0 `. Suppose you approximate ` f(x) ` by ` T_(4)(x) `, find all positive values of x for which this approximation is within 0.001 of the right answer. (Hint: use the...

Homework Statement The Taylor series for f(x) = ln(sec(x)) at a = 0 is sum_(n=0to infinity) c(sub n) (x)^n. Find the first few coefficients. The Attempt at a Solution I've been trying to figure out where to start by looking it up...I've seen instructions that each coefficient is...

This is for revision purposes (not homework so I am not trying to cheat my way out of it!) and its too late in the week to see my lecturer about this. I don't have much of an attempt at the solution because i haven't got a clue where to start. It looks like just a short one though. Here goes...

Homework Statement im being asked for the first 4 non zero values for the taylor expansion of exp(x) which is simple, but then it asks for the range of x values that are valid for the expansion. i have never come across ths before - any idea?

can anybody help me with this integration? Integral of e to the -2x times x squared dx it expands to 1/4, but I'm not sure how to start.

I need to find the Taylor polynomial of degree 4 expanded about a=4 for the function f(x)=squareroot of (x)=x^(1/2) This is what I've started with but I'm not sure how to proceed and if I even started correctly: f'(x)(-1/2)x^(-1/2)=1/2sqrt(x) f"(x)=(-1/4)x^(-3/2)=-1/4x^3/2...

Homework Statement Calculate the taylor polynom of order 3 at (0,0,0) of the function with well-known series (that means I can't just take the derivatives) f(x,y,z)=\sqrt{e^{-x}+\sin y+z^{2}} Homework Equations The Attempt at a Solution I wrote the functions within the square...