taylor

1. Estimating integral using Taylor Series, extra term

Hi I have a question to find the Taylor Series for y(x)=ln(1+x) up to and including terms of 0(x^5). I found that series to be: x - 1/2x^(2) + 1/3x^(3) - 1/4x^(4) + 1/5x^(5)...which may or may not be correct! The next part of the question asks: "Use the series to estimate the value of the...
2. Taylor series for cos(z) about z=pi/4 using Maclaurin series

I want to express $\cos(z)$ in a Taylor series centred on $z_0=\frac{\pi}{4}$. Using the formula $\sum_{k=0}^{\infty} \dfrac{f^{(k)}(z_0)z^k}{k!}$ and the fact that $f^{(k)}(z)=\cos(z+\frac{k\pi}{2})$ in this case I found that the Taylor series is \sum_{k=0}^{\infty}...
3. How to solver this Taylor series?

1. We have function and and we have to find Taylor series for at and find the derivative at .
4. Taylor Series

Use the definition to find the Taylor series (centered at c) for the function. f(x) = cos x, c = (pi/4) I have this one 3 times but to no avail.
5. Taylor Series Expansion?

Hi. I'm reading this one book about pulsatile flow and there's one passage in it that me and my friends can't make sense of (Headbang). You don't need much insight into the physical problem to understand it, so I figured I'd ask for help here. Also, the author probably used a Taylor series...
6. Taylor of [ ln (1+x) ]^2. Help please.

How i can calculate this Taylor polinomy in a=0 ?? Some clue please.
7. Taylor series and Geometric series

To find the taylor series of a function you would usually use the formula $\sum_{n=0}^{\infty}\frac{f^{n}(c)}{n!}(z-c)^n$. However when computing the taylor series for $f(z)=\frac{1}{z+3}$ about $z=1$, I discovered that not only can you compute it using the above formula but you can also...
8. Taylor polynomial

f(x) is differentiable infinite times in R, and there exist L such that |f[n](x)| <= L ( where f[n] means the derivative of f of order n ) for every x and for every natural number n. How do you prove: if f(1/n) = 0 for every natural number n then f(x) = 0 for every x? Thanks.
9. finding derivatives when calculating taylor polynomial

Hi to calculate the taylor polynomial at T3(x) at 1 for f(x) =\frac{1}{x+2} I am having trouble finding the 3rd derivative. Dealing with a rational function so we use the quotient rule, 1st derivative =\frac{0.(x+2)-1(1)}{(x+2)^2}= \frac{-1}{(x+2)^2)} 2nd derivative (this is where I have seem...
10. Finding taylor series

I'm pretty sure I did this right. Just making sure.
11. Taylor Approximations help

I need help with the first part of the problem discussed in this old thread http://mathhelpforum.com/calculus/178346-taylor-s-theorem-question.html I know the formula, but I'm not sure how to find k because I don't know what the actual function is. I know I need to use the sixth derivative. Any...
12. I dont understand my book... convergence of taylor series

I will upload the questions and solutions for 2 problems (35 and 36) of my textbook. I'm confused about the value of M. From Taylor Theorem, you can assume f^{(n+1)} (c) < M for and if lim_{n\to\infty} \frac{Mx^{n+1}}{(n+1)!} \Rightarrow 0 then you can say the entire Taylor Series also...
13. Taylor's and Maclauren's questions

I was following the attached exercises, and I have the following questions: 1) With regards to the first example, why was the 2 included at the beginning of the function? 2) With regards to the second example, why is the last order written as (-1)n (x2n+1)/(2n+1)! and not as...
14. Taylor Series Approximation

Question Determine the function value and the first and second derivative of f (x) = x4 at x = 3. Then, estimate the function value at x = 3.1, using a first-order and second-order Taylor series approximation. How big is the error? Illustrate with a graph. >>Not really sure if im on the right...
15. What is the idea behind defining taylor series in terms of derivatives.

So the basic idea when coming up with a polynomial approximation for a function f(x) at some x = a, is to define a polynomial that has the same value as the function at a. The same value for first derivative at x = a as the value of the first derivative of the function, so on. That is if f(x) is...
16. Convergence of taylor series and function approximation

If a function f(x) is infinitely differentiable on some interval I, you can generate a taylor series for it. This taylor series may or may not converge to f(x). My questions are... If the taylor series generated for f(x) does not converge to f(x) for any x, what does this imply, does that mean...
17. Real analysis and Taylor Series

Hi guys, I've been working on this question since last night and I've been trying to follow the hint and look at the derivative of log(1-z) and the integrate that, but all attempts have been futile. Really lost here... Prove that the Taylor series about the origin of the function [Log(1-z)]²...
18. Taylor expansion??

Hi, me again. I have a question as follows: "Obtain the first four terms of the expansion x around t=1 if x(1) is real and x^3 + 2xt -3 = 0 " I have no idea where to start, please can you help me. I have no proper example in my available text books :( Thank you ever so much Kas
19. Representing functions as summations - Taylor Series

Hello, I have just learned about the Taylor & Maclaurin series and I have a list of questions !!! Why do we represent the functions using polynomial series? Why not any other elementary function? Can all functions be represented as a Taylor-Maclaurin series? What does it mean geometrically...
20. multiplying taylor series

I am trying to multiply the following two taylor series together f(x)=e^-x In(1+x) I know both series expand e.g .In(1+x) =x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4 and e^-x = 1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\frac{1}{4!}x^4 how do I multiply these together , for example do I multiply...