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).
Homework Statement
Find the first 3 non-zero terms of the Taylor polynomial generated by f (x) = x^{3} sin(x) at a = 0.
Homework Equations
f^{n}(x) * (x-a)^{n} / (n!)
The Attempt at a Solution
I got the question wrong: my answer was 1/3! + 1/5! + 1/7!
Here is the answer below. I...
Hi! I'm taking a course on Perturbation theory and as it's quite advanced the lecturer assumes everyone has a good level of maths. One of the parts is expanding roots of a quadratic equation about 0, I can understand how simple ones of the form $(1 + x)^2$ but I don't know where the answers are...
So, I have the series of g(x) = e^{(x-1)^{2}} = 1 + (x-1)^{2} + \frac{(x-1)^{4}}{2} + \frac{(x-1)^{6}}{6} + ... + \frac{(x-1)^{2n}}{n!}
and I am asked to find the series of f(x) = \frac{e^{(x-1)^{2}}-1}{(x-1)^{2}} for x \neq 1 and f(1) = 1. The Taylor series is centered about x = 1
I...
I'm having a hard time understanding the fundamentals of the taylor series. So I get how you continually take derivatives in order to find the coefficients but in order to do that we have to state that x=a. Well when we finally get done we have an infinite polynomial of...
Hi. I just want to ask: how can I realize that I need to do the 4th order taylor's expansion for solving a precise limit? e.g.
\mathop {\lim }\limits_{x\to 0} \frac{{{e^x}-1-\frac{{{x^2}}}{2}+\sin x-2x}}{{1-\cos x-\frac{{{x^2}}}{2}}}
We need the 4th order of the expansion but how can I realize...
1. Solve y'=3t^2y^2 on [0, 3] , y0 = −1, using Euler method and Taylor method of
order 3. Compare your solutions to the exact solution. y(t)=(-1/((t^3)+1))
I DONT KNOW WHAT IS WRONG WITH MY PROGRAM! PLEASE HELP =D
Homework Equations
http://en.wikipedia.org/wiki/Euler_method...
In thermodynamic perturbation theory (chapter 32 in Landau's Statistical Physics) for the Gibbs (= canonical) distribution, we have E = E_0 + V, where V is the perturbation of our energy.
When we want to calculate the free energy, we have:
e^{-F/T} = \int e^{-(E+V)/T} \mathrm{d}\Gamma
We can...
Homework Statement
The first three terms of a Taylor Series centered about 1 for ln(x) is given by:
\frac{x^{3}}{3} - \frac{3x^{2}}{2} + 3x - \frac{11}{6}
and that
\int{ln(x)dx} = xlnx - x + c
Show that an approximation of ln(x) is given by:
\frac{x^3}{12} - \frac{x^2}{2} +...
I've spent all day on this problem and am wasting precious time needed for other work - please give any input you can! The question: given two wages, w1 and w2 where w2 > w1...
a. the difference between the wages as a proportion of the lower: a = (w2 - w1) / w2
b. the difference between the...
Homework Statement
Not much has gotten me in this class, and I almost want to say this has to be a typo, but I want someone else to check it out first.
Homework question is that we need to show that
cos(cos θ)*cosh(sin θ) = Ʃ(-1)ncos(nθ)/(2n)! for n>=0
There is a similar one involving...
Part 1 of Pharaoh's Taylor series and modified Euler question from Yahoo Answers
The Taylor series expansion about \(t=0\) is of the form: \(y(t)=y(0)+y'(0)t+\frac{y''(0)t^2}{2}+.. \)We are given \(y(0)\) and \(y'(0)\) in the initial condition, and so from the equation we have: \(y''(0) =...
Homework Statement
Expand f(x) = x/(x+1) in a taylor series about a=10.
Homework Equations
f(x) = Ʃ (f^n(a)*(x-a)^n / n!
The Attempt at a Solution
I'm having a hard time arriving at the correct answer..I think I'm definitely getting lost somewhere along the way. Here's what I've...
Hi Guys,
Looking at some notes i have on conformal mapping and I have the following
where z is complex and z* denotes its conjugate, R is a real number
z* = -iR + R^2/(z-iR)
and my lecturer says that using the taylor series we get,
z* = -iR + iR(1+ z/iR + ...)
I've been...
I wasn't sure where to put this, so I put this here!
In the photo, you see there's written 'Taylor expanding for small delta-r2, we find' ...
I really don't get the two steps in the next line.
Any help would be greatly appreciated.
The relationship linking the emitted frequency Fe and the received frequency Fr is the Doppler Law:
F_r = \sqrt \frac{1-\frac{v}{c}}{1-\frac{v}{c}} F_e
The Taylor series for the function \sqrt\frac{1+x}{1-x} near x = 0 is 1+x+\frac{x^2}{2}+\frac{x^3}{3}+...
On Earth, most objects travel...
Anyone bored enough to want to help me out with some calculus? I got to deliver this in 6 hours and can't work these out. Help would be SO much appreciated, I've been at it all night and can't make it out.
1. y^2 - e^sin(x) + xy = sin (x)* cos (y) +3
assume y= y(x) and find y ' (0)
2...
If I want to find the taylor series at x = 0 for sin(x^2)+cos(x)...
sin(x^2) = x^2 - x^6/3! + x^10/5! - x^14/7! ...
cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! ...
So why does sin(x^2) + cos(x) = 1 + x^2/2! + x^4/4! + 121x^6/6! ...?
Thanks!
1. Prove that the MacLaurin series for cosx converges to cosx for all x.
Homework Equations
Ʃ(n=0 to infinity) ((-1)^n)(x^2n)/((2n)!) is the MacLaurin series for cosx
|Rn(x)|\leqM*(|x|^(n+1))/((n+1)!) if |f^(n+1)(x)|\leqM
lim(n->infinity)Rn=0 then a function is equal to its Taylor series...
Homework Statement
Hi
Say I want to Taylor-expand
f(\omega + m\sin(\Omega t))
where ω and Ω are frequencies, m is some constant and t denotes time. Then I would get
f(\omega + m\sin(\Omega t)) = f(\omega) + (m\sin(\Omega t)\frac{dI}{d\omega} + \ldots
Is it necessary to make any...
Homework Statement
Find the first two non-zero terms in the Taylor expansion of \frac{x}{\sqrt{x^2-a^2}} where a is a real constantHomework Equations
f(x)=f(x_0)+f^{\prime}(x_0)(x-x_0)+\frac{f^{\prime\prime}(x_0)}{2!}(x-x_0)^2+...+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n
The Attempt at a Solution
If...
This might be a weird quest and in the wrong section but does anyone know of a list with errors and corrections for the book "Advanced Calculus" by Taylor & Mann 3rd ed?
Thanks
Homework Statement
W(x,s)=(1/s)*(sinh(x*s^0.5))/(sinh(s^0.5))
Find the inverse laplace transform of W(x,s), i.e. find w(x,t).
Answer:
w(x,t)= x + Ʃ ((-1)^n)/n) * e^(-t*(n*pi)^2) * sin(n*pi*x)
summing between from n=1 to ∞
Homework Equations
An asymptotic series..?!
The...
i am very confuse how my profs always use taylor expansion in physics which somehow doesn't follow the general equation of
f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)2 and so on...
like for example, what is the taylor expansion of x - kx where k is small
it was given as something like...
Homework Statement
I have to give the range of validity for a Taylor series built from an expression of the form:
(1+(a/b)x)^c
Homework Equations
The Attempt at a Solution
Obviously the validity does not extend to x=-(b/a) on the negative side, but should I then state that...
1. Problem: if f(1,3)=7, use Taylor expansion to describe f(1.2,3.1) and f(.9,2.8) if the partials of f are give by
df/dx=.2
d^2f/dx^2=.6
df/dy=.4
d^2f/dy^2=.9
(you do not need to go beyond the second derivative for this problem)
2. I know from class how to do this if one variable changes...
Homework Statement
The course is Computational Physics, but in a sense this is a pretty straight computer science or even mathematical challenge.
The first part of the assignment - the relatively easy part - was to write a Fortran program to take two variables - the number to which e...
Homework Statement
Here is the question:
I don't quite know what I did wrong. My method is below.
Homework Equations
The Attempt at a Solution
f(x)=√x
f'(x)=\frac{1}{2(x)^{1/2}}
f''(x)=\frac{-1}{(2)(2)(x^{3/2}}
a=4
f(a)=2
f'(a)=1/4...
Use the error estimate for Taylor polynomials to find an n such that
| e - (1 + (1/1!) + (1/2!) + (1/3!) + ... + (1/n!) | < 0.000005
all i have right now is the individual components...
f(x) = ex
Tn (x) = 1/ (n-1)!
k/(n-1)! |x-a|n+1 = 0.000005
a = 0
x = 1
I don't know where to go from here
Not sure under which forum this should have gone under, anyway can someone who really understands it explain it to me in as simple terms as they can, from what I'm getting its approximates something for a function or something? No idea.
Homework Statement
I was given the following problem, but I am having a hard time interpreting what some parts mean.
We're given the equation
sinθ+b(1+cos^2(θ)+cos(θ))=0
Assume that this equation defines θ as a function, θ(b), of b near (0,0). Computer the Taylor polynomial of...
Homework Statement
Derive the following formula using Taylor series and then establish the error terms for each.
Homework Equations
f ' (x) ≈ (1/2*h) [4*f(x + h) - 3*f(x) - f(x+2h)]
The Attempt at a Solution
I honestly have no idea how to go about deriving this. The professor did...
Homework Statement
Hi! I need to solve the following ODE:
xy'=1-y+x^2y^2, \qquad y(0)=1
using a predictor-corrector method. Starting values need to be found using a Taylor method.
The exact solution is of the form \frac{\tan{x}}{x}
Homework Equations
Taylor method of third order...
$1+v_{t+1} = (1+v_t)\exp\left(-rv_{t-1}\right)\approx (1+v_t)(1-rv_{t-1})$
The book is linearizing the model where we generally use a Taylor Series.
How was the expression expanded in the Taylor Series to get the approximate answer?
Thanks.
I am attempting to complete a problem for a problem set and am having difficulty simplifying an expression; any help would be greatly appreciated!
The question is a physics question which attempts to derive an equation for the temperature within a planet as a function of depth assuming...
Homework Statement
Show that if F is twice continuously differentiable on (a,b), then one can write
F(x+h) = F(x) + h F'(x) + \frac{h^2}{2} F''(x) + h^2 \varphi(h),
where \varphi(h) \to 0 as h\to 0.
Homework Equations
The Attempt at a Solution
I'm posting this here...
Bit stuck on this. I tried writing 1/(1-z^2) as taylor series then Cos z as taylor series, then substituting one into the other but it looked a bit dodgy. Can one simple substitute like this?
I have to find the first three non zero terms of this series by hand. I know the answer and it is
-(z^3/3) - z^7/2520 - z^11/19958400
Which will take ages to get to by brute force. Is there a quicker way?
I am trying to find the Taylor series for
$$\displaystyle
\dfrac{\left(\dfrac{1}{z-i}\right)}{z+i}
$$
where z is a complex number.There is a reason it is set up as a fraction over the denominator so let's not move it down.
Hi there,
I was hammering out the coefficients for the Taylor Series expansion of f(x) = \frac{1}{\sqrt{1-x^2}}, which proved to be quite unsatisfying, so decide to have a look around online for alt. approaches.
What I found (in addition to the method that uses the binomial theorem) was...
Homework Statement
The magnitude of the gravitational force exerted by the Earth on an object of mass m at the Earth's surface is
Fg = G*M*m/ R^2
where M and R are the mass and radius of the Earth.
Let's say the object is instead a height y << R above the surface of the Earth. Using a...
Homework Statement
Find the critical point(s) of this function and determine if the function has a maxi-
mum/minimum/neither at the critical point(s) (semi colons start a new row in the matrix)
f(x,y,z) = 1/2 [ x y z ] [3 1 0; 1 4 -1; 0 -1 2] [x;y;z]
Homework Equations
The...
Hello,
I have two functions say f1(β) and f2(β) as follows:
f1(β)=1/(aδ^2) + 1/(bδ) + O(1) ... (1)
and
f2(β)= c+dδ+O(δ^2) ... (2)
where δ = β-η and a,b,c,d and η are constants. Eq. (1) and (2) are the Taylor series expansions of f1(β) and f2(β) about η...
Under what circumstances is it correct to say of the function u(x) \in L^2(-\infty,\infty) that
u(x-t) = u(x) - \frac{du}{dx}t + \frac 12 \frac{d^2u}{dx^2}t^2 - \cdots = \sum_{n=0}^\infty \frac{u^{(n)}(x)}{n!}(-t)^n.
Homework Statement
This is part of a larger problem, but in order to take what I believe is the first step, I need to take the Taylor series expansion of f(x,y) = \cos\sqrt{x+y} about (x,y) = (0,0)
On the other hand, the purpose of doing this expansion is to find an asymptotic expression for...
Hi,
I have a following question...
Can it be that there is given some Lagrangian and instead of considering whole Lagrangian one makes its series expansion and considers only some orders of expansion? Can you bring some examples or why and when does this happen... ?
Thank you
Hello all,
My question is in regards to the Taylor series expansion of
f(x)=e^x=1+x+x^2/(2!)+x^3/(3!)...
I calculated the value of
e^(-2)
using the first 4 terms, 6 terms, and then the first 8 terms. I then calculated the relative error to compare it to the true value, depcited by my...
Homework Statement
Hi!
I have a couple of problems on Taylor Series Approximation.
For the following equations, write out the second-order Taylor‐series approximation.
Let x*=1 and, for x=2, calculate the true value of the function and the approximate value given by the Taylor series...
http://uploadpic.org/storage/2011/dGTvcFGgvl4VYoVB40HpLSMxH.jpeg
Can somebody guide me through this? I know how to apply Newton Raphson Method, but the x^* symbol and "argmin" function are kinda new to me. I am re referring to part (c). Thanks.
Homework Statement
Taylor's theorem can be stated f(a+x)=f(a)+xf'(a)+(1/2!)(x^2)f''(a)+...+(1/n!)(x^n)Rn
where Rn=fn(a+y), 0≤y≤x
Use this form of Taylor's theorem to find an expansion of sin(a+x) in powers of x, and show that in this case, mod(\frac{x^n Rn}{n!})\rightarrow0 as...