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).
I was just pondering today how the kinematic equation for position looks like a taylor expansion.
x = x0 + dx/dt *t + (1/2)*d2x/dt2*t2
I believe it continues like that, exactly like a taylor expansion does, so the next term would be (1/6)*d3x/dt3*t3
If it is indeed a taylor expansion, what...
Homework Statement
Let f be differentiable on [a,b] and f'(a)=f'(b)=0. Prove that if f'' exists then there exists a point c in (a,b) such that
test
|f''(c)| \geq \frac{4}{(b-a)^2}|f(b)-f(a)|
Homework Equations
All of the equations are supposed to be in absolute value but I had...
Hello, I was wondering if anyone could explain to me the thought process behind how you find the maximum remainder of a Taylor series?
I read the wiki article and didn't help me at all,
http://en.wikipedia.org/wiki/Taylor's_theorem
My book talks about something like this(image is...
Homework Statement
For f(x) = xln(x), find the taylor series expansion of f(x) about x = 1, and write the infinite series in compact form.
2. The attempt at a solution
I can find the expansion itself fine, these are the first few terms:
0 + (x-1) + \frac{(x-1)^{2}}{2!} -...
Homework Statement
a) Using a geometric series, find the Taylor expansion of the function f(x) = x/(1+x^2)
b) Use the series found in (a) to obtain the Taylor expansion of ln(1 + x^2)
Homework Equations
The Attempt at a Solution
I really don't know where to start; I can't find...
Hi everyone. The problem I have to face is to perform a taylor series expansion of the integral
\int_{-\infty}^{\infty}\frac{e^{-\sum_{i}\frac{x_{i}^{2}}{2\epsilon}}}{\sqrt{2\pi\epsilon}^{N}}\cdot e^{f(\{x\})}dx_{i}\ldots dx_{N}
with respect to variance \epsilon. I find some difficulties...
so F = mgR2/(R+h)2
where R is the radius of the earth. consider the situation where h is much smaller than R.
a) show that F is approximately equal to mg
b)express F as mg multiplied by a series in h/R
so i need help on getting started.
would showing that F is approximately equal...
Homework Statement
find the taylor series for the function
f(x) = \frac{x^2+1}{4x+5}
Homework Equations
N/A
The Attempt at a Solution
how to do this?
1st attempt.
i did turn it this term
\frac{x}{4} + \frac{-5x+4}{16x+20} can i turn this to taylor series?
maybe i know how to make...
I don't have anyone else to ask. So I have to ask you guys. I learned about Taylor series, and then I went back and looked at linear and quadratic approximations, and they are Taylor series except only taken so far. I'm pretty much just looking for confirmation on my idea, it seems perfect.
Homework Statement
find the taylor polynomial f4 for f(x)=sin(2x) and a=pi/4
Homework Equations
sin(x)=((-1)^n)(x^(2n+1))/((2n+1)!)
The Attempt at a Solution
so replace x with 2x?
you get ((-1)^n)(2x)^(2n+1)/(2n+1)!)
is this right?
Homework Statement
find taylor series for \frac{x-1}{1+x} at x=1
Homework Equations
The Attempt at a Solution
how to change this form
\frac{x-1}{1+x}
to something like this
\frac{1}{1+a} or \frac{1}{1-a}
help me please T_T
or should i do like this
\sum\frac{f^n(1)(x-1)^n}{n!}
and find...
Homework Statement
Find the Taylor Series for f(w) = 1/w centered at w0 = 1 using 1/w = (1/1 + (w-1)). Show that the series converges when |w-1| < 1
Homework Equations
use 1/w = (1/1 + (w-1))
The Attempt at a Solution
I have two equations:
\ddot{x}^\mu + \ddot{y}^\mu + \Gamma^\mu{}_{\nu \lambda} (x+y)(\dot{x}^\nu+\dot{y}^\nu)(\dot{x}^\lambda+\dot{y}^\lambda)=0
and
\ddot{x}^\mu + \Gamma^\mu{}_{\nu\lambda}(x) \dot{x}^\nu \dot{x}^\lambda=0
apparently if i taylor expand the first equation to first order...
Homework Statement
Use the taylor's expansion of f(x)= x1/4 about x= 16 to estimate (16.1)1/4
Homework Equations
Taylors formula: f(a) + f'(a) (x-a) + (f''(a)/2!) (x-a)2+...The Attempt at a Solution
Ok I have calculate the taylor expansion to be: 2 + (1/32) (x-16)-(3/320) (x-16)2+ (7/262144)...
Homework Statement
Derive a method for approximating f'''(x0) whose error term is of order h^{2} by expanding the function f in a fourth taylor polynomial about x0 and evaluating at x_{0} \pm h and x_{0} \pm 2h.
Homework Equations
The Attempt at a Solution
I'm not sure where to...
Homework Statement
This is a three part question: It is based off the first two sections. I'm pretty sure the first two answers are correct, but I have no idea how to do the third question.
Write the First three nonzero terms and the general term of the Taylor series expansion about x=0...
Homework Statement
If \sum_{n=0}^{\infty} a_{n}x^n is a Taylor series that converges to f(x) for all real x, then f'(1) = ?
Homework Equations
A Taylor series:
\sum_{n=0}^{\infty} \frac {f^{(n)}(c)}{n!}(x-c)^n
and the dirv of a Taylor series:
f'(x)=\sum_{n=0}^{\infty}...
Homework Statement
The question asks me to write out a taylor polynomial for 1/(1-x^2)
of degree 2n+1 at 0.
The Attempt at a Solution
My answer was 1 + x^2 + x^4 + x^6 + ... + (x^4)/(1-x^2) which I just got from using hte geometric series formula. The textbook answer however...
Homework Statement
Determine the order two Taylor polynomial, p2(x, y), for
f(x, y) = log e (1 + x2 + y4)
about point (0, 1)
ANSWER:
loge (2) + 2y - 2 + \frac{1}{2} [ x2 - 2y2 + 4y - 2 ]
Managed that question and should be correct. If not, do let me know =)
Part 2: Using...
Homework Statement
Write the Taylor series of the function f(x) = (\pi -x)^-2 around a = 0
Homework Equations
(\pi - x)^-2 = f(a) + f'(a)(x-a) + [f''(a)(x-a)^2]/(2!) +...+ [f^n(a)(x-a)^n]/(n!)
The Attempt at a Solution
This is what i have and i am not sure i am showing it...
Homework Statement
Write a user-defined function that determines cos(x) using Taylor Series expansion
Stop adding terms when estimated error, E<=.000001
Homework Equations
sum Sn = Sn-1 + an
E = | (Sn - Sn-1)/Sn-1 |
The Attempt at a Solution
function y = cosTaylor(x)
Sn=1...
Homework Statement
Solve the differential equation
\frac{dy^2}{dx^2}=xy^2-2yy'+x^3+4
where
y(1)=1
y'(1)=2
by means of the Taylor-series expansion to get the value of y at x=1.1. Use terms up to x^6 and \Delta x=0.1The Attempt at a Solution
I'm unsure as to how I should go about...
Let f be the function given by f(t) = 4/ (1 + t^2) and G be the function given by G(x) = {Integral from 0 to x} f(t)dt .
(a) Find the first four nonzero terms and the general term for the power series expansion of f(t) about t = 0.
(b) Find the first four nonzero terms and the general term...
Homework Statement
Let f(x) = sin x
a) find p_6 (taylor polynomial 6th degree) for f at x = 0
b) How accurate is this on the interval [-1,1]
Homework Equations
The Attempt at a Solution
I got p_6 = x + (x^3)/6 + (x^5)/120, which was correct as per the solution manual. My...
I just need help on how to start the problem, I'm not asking anyone to do it for me, I'm just slightly confused.
What is the degree of the Taylor polynomial needed to approximate sqrt(e) with error < 0.001. Use ex as your function, with x = 0.5.
I'm just honestly confused on where to even...
Homework Statement
I am trying to find the Tn(x) for sqrt[x] centered at a=1
Homework Equations
The Attempt at a Solution
right now i have
f'(x)=1/2x^-1/2
f''(x)=-1/4x^-3/2
f'''(x)=3/8x^-5/2
f''''(x)=-15/16x^-7/2
f'(1)=1/2
f''(1)=-1/4
f'''(1)=3/8
f''''(1)=-15/16
how...
Homework Statement
Find the taylor expansion of the following formula in the case where r > > d to the first order in \epsilon = \frac{d}{r}
\frac{1}{r_{+}} = \frac{1}{\sqrt{r^{2} + (\frac{d}{2})^{2} - rdcos\theta}}
Homework Equations
(1 + \epsilon)^{m} = 1+m\epsilon, where...
Homework Statement
I understand the whole concept of Taylor Series and Maclaurin series but I don't know how to rewrite them in sigma notation.
I'll use this generic example. Find the Maclaurin series of the function \ f(x)=e^{x}
Homework Equations
The Attempt at a Solution
\...
I was hoping somebody would be able to help me as I am pretty new to Matlab. I am trying to create a for-loop to describe the taylor series expansion of cos(x)= (-1)^n*x^2n/(2n)! and to see how it converges towards cos(x). Below is the code that I have used to plot the different orders of n, but...
Homework Statement
Using the technique of Taylor expansion, find an approximate expression for the relativistic factor γ for small v (i.e., expanded around v = 0) that is correct to order v2.
Homework Equations
γ=1/SQRT(1+ V2/C2). But in class, my professor just substituted X=V/C, so...
when i develop the series of a cosine i have a (-1) member
i wanted to represent the series as a sum
so i need to take only the odd members so the power of -1 is 2k+1 i got
but the solution says that the power of -1 is equal (-1)^{k-1}
is it the same??
why they have such an expression...
Hello,
Is there any place I can find the equation for the Taylor expansion of a functional around a function ??
Particularly, I want something like:
f[x(t)] = f[\hat{x}(t)] + (f[\hat{x}(t)] - f[x(t)] \frac{\delta f}{\delta x(t)}|_{x(t)=\hat{x}(t)} + \frac{(f[\hat{x}(t)] -...
I'm currently studying the Taylor series and I cannot figure out how the remainder term came to be. If anyone could clarify this for me, I would be really grateful ...!
I understand that the Taylor series isn't always equal to f(x) for each x, so we put Rn at the end as the remainder term...
[URGENT] Taylor Series without using the built-in MATLAB "Taylor's Function"
I have a MATLAB Test Tomorrow
Please teach me the MATLAB programming to solve Taylor & Maclaurin Series, without using the built-in MATLAB "Taylor's Function"
Please explain the procedure to solve them using the...
Homework Statement
Find the taylor series of f(x)=1/(x)^(1/2) ; a=9
2. The attempt at a solution
f(x) = (x)^(-1/2)
f'(x) = -(1/2)*x^(-3/2)
f''(x) = (1/2)*(3/2)*x^(-5/2)
f'''(x) = -(1/2)*(3/2)*(5/2)*x^(-7/2)
f''''(x) = (1/2)*(3/2)*(5/2)*(7/2)*x^(-11/2)
f(9) =...
Trying to find the Taylor Series for cos(x) where x0 is PI.
I've gotten
cos(x) -1
-sin(x) 0
-cos(x) 1
sin(x) 0
cos(x) -1
It's clearly 0 every other term so I need 2k or 2k-1. But the -1 term switches between -1 and 1
How in world do I deal with this? xD
Thanks for any...
The series is:
(33/5) - (34/7) + (35/9) - (36/11)+...
Looking at this, I'm guessing I can use the Taylor Series for arctan(x) but I don't know how to apply it or where to begin.
Any help is greatly appreciated.
The Taylor Series of sin(x)=x-(x3/3!)+(x5/5!)-...
What function of sin gives the following:
(\pi2/(22) - (\pi4/(24*3!)+ (\pi6/(26*5!) - (\pi8/(28*7!)+...
Please help me.
Thank you.
Homework Statement
What function produces the following:
(\pi2/(22)) - (\pi4/(24*3!)) + (\pi6/(26*5!)) - (\pi8/(28*7!))
I'm sure this is a sin function.
But I can't figure out what exactly is the function.
Please help.
Homework Statement
Use taylor series method to compute the integral from 1 to 2 of [sin(x2)] / (x2) with 10 -3 precision
Homework Equations
The Attempt at a Solution
I'm not sure where to start. Someone please help me.
Homework Statement
Find a taylor series for f(x)=sq. rt. of X about c=1
Homework Equations
N/A
The Attempt at a Solution
I took the derivative of the sq rt of X, and then plugged in 1 for all the X's. I got:
f(x)= 1
f'(x)=1/2
f''(x)=-1/4
f'''(x)=3/8
f^4(x)=-15/16
My teacher...
Homework Statement
For what values of x do you expect the following Taylor series to converge?
sqrt(x^{2}-x-2)
Homework Equations
I'm not too sure
The Attempt at a Solution
Well quite frankly I have no idea what to do. If someone can push me in the right direction I'll get the rest done.
Homework Statement
For what values of x (or \theta or u as appropriate) do you expect the following Taylor Series to converge? DO NOT work out the series.
\sqrt{x^{2}-x-2} about x = 1/3
sin(1-\theta^{2}) about \theta = 0
tanh (u) about u =1
Homework Equations
The...
Homework Statement
What degree Taylor Polynomial around a = 0(MacLaurin) is needed to approximate cos(0.25) to 5 decimals of accuracy?
Homework Equations
taylor series...to complicated to type out here
remainder of nth degree taylor polynomial = |R(x)| <= M/(n+1)! * |x - a|^(n+1)...
Homework Statement
f(x) = \frac{ln(3x)}{6x}, a = \frac{1}{3}, n=3
Find T3
Homework Equations
Taylor Series - f(n)(x)/n! * (x-a)^n
The Attempt at a Solution
So, I isolated ln(3x) from 1/6x.
I created the series based off of ln(3x).
f(0)(x)=ln(3x) ->f(0)(1/3)=ln(3(1/3)) =0...
https://www.amazon.com/dp/189138922X/?tag=pfamazon01-20
Has anyone ever read this book? It looks like a bargain, good reviews, low price. What do you think of it? Is it a good mathematically oriented physics book?
Is a Fourier series essentially the analogue to a Taylor series except expressing a function as trigs functions rather than as polynomials? Like the Taylor series, is it ok only for analytic functions, i.e. the remainder term goes to zero as n->infinity?