In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.
If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).
Hello,
I am reading a course on signal processing involving the Z-transform, and I just read something that leaves me confused.
Let ##F(z)## be the given Z-transform of a numerical function ##f[n]## (discrete amplitudes, discrete variable), which has a positive semi-finite support and finite...
This attractor is unusual because it uses both the tanh() and abs() functions. A picture can be found here (penultimate image). Here is some dependency-free Python (abridged from the GitHub code, but not flattened!) to generate the data to arbitrary order:
#!/usr/bin/env python3
from sys...
I came across this basic limits question
Ltx->0[(ln(1+X)-sin(X)+X2/2]/[Xtan(X)Sin(X)]
The part before '/'(the one separated by ][ is numerator and the one after that is denominator
The problem is if I substitute standard limits :
(Ltx->0tan(X)/x=1
Ltx->0sin(X)/X=1
Ltx->0ln(1+X)/X=1)
The...
I have written some ODE solvers, using a method which may not be well known to many. This is my attempt to explain my implementation of the method as simply as possible, but I would appreciate review and corrections.
At various points the text mentions Taylor Series recurrences, which I only...
Homework Statement
Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n##
Evaluate: ##f^{(8)}(4)##
Homework Equations
The Taylor Series Equation
The Attempt at a Solution
Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...
Homework Statement
Show that the magnitude of the net force exerted on one dipole by the other dipole is given approximately by:$$F_{net}≈\frac {6q^2s^2k} {r^4}$$
for ##r\gg s##, where r is the distance from one dipole to the other dipole, s is the distance across one dipole. (Both dipoles are...
We were informally introduced Taylor series in my physics class as a method to give an equation of the electric field at a point far away from a dipole (both dipole and point are aligned on an axis). Basically for the electric field: $$\vec E_{axis}=\frac q {4πε_o}[\frac {1} {(x-\frac s 2)^2}-...
I've read that, in general, the energy of a wave, as opposed to what's commonly taught, isn't strictly related to the square of the amplitude. It can be seen to be related to a Taylor series, where E = ao + a1 A + a2A2 ... Also, that the energy doesn't depend on phase, so only even terms will...
I already learn to use Taylor series as:
f(x) = ∑ fn(x0) / n! (x-x0)n
But i don´t see why the serie change when we use differents x0 points.
Por example:
f(x) = x2
to express Taylor series in x0 = 0
f(x) = f(0) + f(0) (x-0) + ... = 0 due to f(0) = (0)2
to x0=1 the series are...
Homework Statement
Find the Taylor series for:
ln[(x - h2) / (x + h2)]
Homework Equations
f(x+h) =∑nk=0 f(k)(x) * hk / k! + En + 1
where En + 1 = f(n + 1)(ξ) * hn + 1 / (n + 1)!
The Attempt at a Solution
ln[(x - h2) / (x + h2)] = ln(x-h2) - ln(x + h2)
This is as far as I have been able to...
Hi, I've got this:
$$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close).
Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}##
But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...
Homework Statement
Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression:
$$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$
Homework Equations
3. The Attempt at a Solution [/B]
##L=\lim_{x \rightarrow 0}...
Homework Statement
Using Taylor series, Find a polynomial p(x) of minimal degree that will approximate F(x) throughout the given interval with an error of magnitude less than 10-4
F(x) = ∫0x sin(t^2)dt
Homework Equations
Rn = f(n+1)(z)|x-a|(n+1)/(n+1)![/B]
The Attempt at a Solution
I am...
I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how.
$$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$
I get most of the function, I just can't see where the ##-1## comes from. Could...
I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector.
Does that mean I will have to treat it as a Taylor expansion about two variables...
Homework Statement
Find a power series that represents $$ \frac{x}{(1+4x)^2}$$
Homework Equations
$$ \sum c_n (x-a)^n $$
The Attempt at a Solution
$$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$
since \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2}
$$ x*\frac{d}{dx}\frac{1}{(1+4x)^2}...
I studied Taylor series but I would like to have an answer to a doubt that I have. Suppose I have ##f(x)=e^{-x}##. Sometimes I've heard things like: "the exponential curve can be locally approximated by a line, furthermore in this particular region it is not very sharp so the approximation is...
Homework Statement
Find the Taylor Series for f(x)=1/x about a center of 3.
Homework Equations
The Attempt at a Solution
f'(x)=-x^-2
f''(x)=2x^-3
f'''(x)=-6x^-4
f''''(x)=24x^-5
...
f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n
I'm not sure where I went wrong...
Homework Statement
\lim\limits_{x \to 0} \left(\ln(1+x)\right)^x
Homework Equations
Maclaurin series:
\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ...
The Attempt at a Solution
We're considering vanishingly small x, so just taking the first term in the...
So, I was doing a question on Laurent series. Part of it asked me to work out the pole of the function:
$$ exp \bigg[\frac{1}{z-1}\bigg]$$
The answer is ##1## - since, we can write out a Maclaurin expansion:
(1) $$ exp\bigg[\frac{1}{z-1}\bigg] = 1+\frac{1}{z-1}+\frac{1}{2!}\frac{1}{(z-1)^{2}}...
Homework Statement
## L (v^2 + 2 \pmb{v} \cdot \pmb{ \epsilon } ~ + \pmb{ \epsilon} ^2)##, where ## \pmb{\epsilon}## is infinitesimal and ##\pmb{v}## is a constant vector (## v^2 ## here means ## \pmb{v} \cdot \pmb{v} ## ), must be expanded in terms of powers of ## \pmb{\epsilon} ## to give...
Hello,
I want to prove that the taylor expansion of f(x)={\frac{1}{\sqrt{1-x}}} converges to ƒ for -1<x<1. If I didn't make a mistake the maclaurin series should look like this:
Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n
My attempt is to use the lagrange error bound, which is...
Homework Statement
To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do with...
In my multivariable calculus class, we briefly went over Taylor polynomial approximations for functions of two variables. My professor said that the second degree terms include any of the following:
$$x^2, y^2, xy$$
What surprised me was the fact that xy was listed as a nonlinear term.
In...
I don't think I've fully grasped the underlying ideas of this class, so at the moment I'm just sort of flailing for equations to plug stuff into...
Homework Statement
Show that in the mean field model, M is proportional to H1/3 at T=Tc and that at H=0, M is proportional to (Tc - T)1/2...
Homework Statement
For ln(.8) estimate the number of terms needed in a Taylor polynomial to guarantee an accuracy of 10^-10 using the Taylor inequality theorem.
Homework Equations
|Rn(x)|<[M(|x-a|)^n+1]/(n+1)! for |x-a|<d.
The Attempt at a Solution
All I've done so far is take a couple...
I am studying power series right now and I am understanding well how to write them and where they converge but I am having some trouble grasping the Taylor Remainder Theorem for a few reasons.
First of all it says the remainder is:
f^(n+1)(c)(x-a)^(n+1)/(n+1)! for some c between a and x.
I...
I am just trying to clarify this point which I am unsure about:
If I am asked to write out (for example) a third order taylor polynomial for sin(x), does that mean I would write out 3 terms of the series OR to the x^3 term.
x-x^3/3!+x^5/5!
or just
x-x^3/3!
Also, I have a question for the...
The problem is as the title says. This is an example we went through during the lecture and therefore I have the solution. However there is a particular step in the solution which I do not understand.
Using the Taylor series we will write sin(x) as:
sin(x) = x - (x^3)/6 + (x^5)B(x)
and...