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
Compute the 9th derivative of
f(x) = \frac{\cos\left(3 x^{4} \right) - 1}{x^{7}}
at x=(0)
Homework Equations
f(x)=\sum^{\infty}_{n=0} \frac{f^{(n)}(c)}{n!}x^n
cos(x)=\sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n)!}x^{2n}
The Attempt at a Solution
The correct answer is...
Homework Statement
Write the Machlaurin series for :
f(x) = (2x)/(1+x2)
Homework Equations
The Attempt at a Solution
I tried finding all the derivatives (aka f(x), f'(x), f''(x), etc..) but the equations started getting longer and longer and would always result in 0 when x=0...
Since ln(0) doesn't exist, this question is futile right?
I am tasked with finding a Maclaurin powerseries for ln(x) and to find out how many times I have to run that series to get a accurate answer for ln(1.5).
What should I do? Should I find the taylor series for ln(1.5) for should I...
Homework Statement
f(x) = e^(x^2) * sin(x)
Find the value of the 3rd derivative at x = 0.
Homework Equations
e^x = 1 + x + x^2/2! + ... + x^n/n!
sin(x) = 1 + x^3/3! - x^5/5! + ... + x^(2n+1)/(2n+1)! * (-1)^(n-1)
The Attempt at a Solution
I know I should plug in the two...
Homework Statement
How can I calculate it for \frac{1}{1+cos^2(x)} by using the fact that \frac{1}{1+x^2} = 1 - x^2 + x^4 - ...?
Homework Equations
Given in the problem.
The Attempt at a Solution
I tried letting u = cos(x), then
\frac{1}{1+cos^2(x)} = \frac{1}{1+u^2} = 1...
Homework Statement
I got a few functions I need to expand to series using Maclaurin forumlas.
Homework Equations
http://mathworld.wolfram.com/MaclaurinSeries.html
The Attempt at a Solution
So here are the ones I managed to do:
f= \sqrt{1-x^2-y^2}
writing it in another form...
Homework Statement
Wikipedia states that the Maclaurin Series expansion of the Lorentz factor is http://en.wikipedia.org/wiki/Lorentz_factor"
Homework Equations
Relevant equations are all found in that article
The Attempt at a Solution
I don't see how this comes about. My attempt...
Homework Statement
find coefficient of x^4 in the MAclaurin series for f(x)=e^sinx
Homework Equations
ok... so taking derivatives 4 times for this function...gave me a mess! @.@
can someone help me in simplying the derivatives...?
1. cosxe^sinx
then for 2. is it...
Homework Statement
The Maclaurin series for a function f is given by \sum\frac{x^n}{2n}. What is the value of f(4)(0), the fourth derivative of at x = 0?
a.) 1
b.) 2
c.) 3
d.) 4
e.) 5
Homework Equations
The Maclaurin Series is the infinite series centered at x = 0 with the following...
Homework Statement
Find the first six nonvanishing terms in the Maclaurin series solution of the initial value problem (x^2 - 3)y''(x) + 2xy'(x) = 0 where y(0) = y0 and y'(0) = y1.
Homework Equations
The Attempt at a Solution
Should with just something like Φ(x) such that Φ(x) =...
Homework Statement
Find the Maclaurin Series for g(x)= (4)/(4+2x+x^2) and its interval of convergence.
Homework Equations
I know the Maclaurin Series usually involves taking derivatives but every other problem I've done so far has had a degree that I've solved to. So, other than the general...
Homework Statement
Find the Maclaurin series for f(x)= 1/ (1+x+X2)
Homework Equations
The Attempt at a Solution
I think the book says I can just divide 1 by the Maclaurin series of (1+x+X2). And when i do this the original function is the answer (which makes sense).
But when...
Homework Statement
(1 pt) Compute the 9th derivative of:
f(x) = \frac{cos(6x^4)-1}{x^7}
at x=0.
Homework Equations
Hint: Use the MacLaurin series for f(x).
The Attempt at a Solution
I have tried many weird ways and cannot come up with the correct numerical answer. I've...
Find the first few terms of the Taylor Series around x=0...?
of the function
f(x)= {x/(e^x - 1) , x =/ 0}
{1 , x=0}
the function is piecewise.
up to and including the term involving x^2
It says to not compute derivatives of f but to use the formula for the...
Homework Statement
5) For each of the series below, write the series in summation notation and give the first five terms of the series. Also give the radius of convergence of the series.
a) Use the series for \frac{1}{1 - x} to find the Maclaurin series of
f(x) = \frac{1}{(1-2x)^3}...
Hey, here's is my problem as the exam states it:
A) Write out the first four non-zero terms of the Maclaurin Series for F(x) = (1+X^{7})^{-4} Give all of the coefficients in exact form, simplified as much as possible.
B) Find the exact value of the 21st order derivative of F(x) =...
Hi all,
I am trying to work out a series expansion for ln ((x+1)/(x-1)).
I have got the series expansion for ln(x+1) ie x- (x^2/2) + (x^3/3) - (x^4/4) ...
and for ln(x-1) -x- (x^2/2) - (x^3/3) - (x^4/4) ...
Can I tie these two together to get the series for ln...
I have the following problem:
find the first 3 non-zero terms in the Maclaurin series for the function:
e-x2 + Cos[x]
I know in this case, the series behave like polynomials and I have done the following. The left expression is the first 3 terms of the e portion of the problem, and the...
Okay I was given this problem as a challenge question. It simply says expressing 2x power as a Maclaurin Series. At first, following an example given by my instructor, I thought that by examining the function as I took multiple derivatives I could find a pattern.
By as you can imagine taking...
Homework Statement
find the maclaurin series of e^(3x) + e(-3x)
Homework Equations
The Attempt at a Solution
I'm not sure about finding taylor and maclaurin series, I understand perfectly how to find the terms of the series...But how do I put it into a general term do I just...
Homework Statement
find the Maclaurin Series:
f(x)=(e^x - cos(x))/x
Homework Equations
The Attempt at a Solution
I'm not really sure what to do I do know the Maclaurin series of cos(x) I was thinking that using this known series then doing e^x - the known series then...
Homework Statement
Show that the function defined by
(stepwise)
f(x)= e^(-1/x^2) if x =/ 0
= 0 if x=0
is NOT equal to its Maclauren series.
Then graph the function and comment on its behavior near the origin.The Attempt at a Solution
Well, I honestly don't know how to...
Today, we're taught that MacLaurin series is just another name for Taylor series at x = 0. Then what is the speciality of it? Why doesn't x = 1 or x = 2 have a special name?
Find the Maclaurin series for f(x) = (x^2)(e^x)
the book suggests obtaining the Maclaurin series of f(x) by multiplying the known Maclaurin series for e^x by x^2:
(x^2)(e^x) = (x^2)(1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...)
= x^2 + x^3 + (x^4)/2! + (x^5)/3! + (x^6)/4! + ... =...
Hey all. We never covered the Maclaurin series (covered Taylor) but these questions are on my review for my final.
Homework Statement
Compute the 9th derivative of:
arctan((x^3)/2) at x=0
f^9(0)=?
Homework Equations
Use the MacLaurin series for f(x)
The Attempt at a Solution...
Find the MacLaurin polynomial of degree 4 for f(x)
f(x)= (integral from 0 to x) sin(3t^2)
[f^(n)(0)/n!]*x^n
The Attempt at a Solution
- I took the 4th derivative of sin(3t^2) and got:
f''''(x)= -108sin(3t^2)-1296t^2cos(3t^2)+1296t^4sin(3t^2)
Not real sure what to do from...
For the function f(x) = (10x^2) e^(-2x), I calculated the first term of the Maclaurin series to be 0.
However, for the second term, I also calculated it to be 0, but apparently this is wrong. Shouldn't the second term be f'(x) = 10x^2 * -2e^(-2x) + e^(-2x) * 20x? Or what am I doing...
Im really having trouble wrapping my mind around the topic of maclaurin series, my problem requires me to find the maclaurin series of f(x) = 1/(1+x^2), then use that to find the maclaurin series of g(x)= arctan(x)...i don't even know where to begin.
I was asked to find dy/dx of inverse tan of x , which is 1/(1+x^2)
Then its says, using that dy/dx ^^^^^ equal to a particular series, find the first 4 terms of inverse tan of x.
I'm confused? What is it asking here??
Homework Statement
find a_{n} for f(x)\ =\ \frac{1}{1+3x}
The Attempt at a Solution
I got:
f(0) = 1
f'(0) = -3
f''(0) = 9
The answer I ended up with was:
a_{n} \ = \ {(-1)}^n\frac{3^n}{n!}
However, the answer in the back of my book has the same answer except it's not divided by n!.
Please Refer to the attachment, I'm new to the forums and am not sure how to type Math Terms.
I've been working on this for a couple of days now and i can't get anywhere near an answer.
Any Help is MUCH appreciated! Thanks!
I have been asked to differentiate cos x and six...the maclaurin series versions...
I have done the general and specific terms as shown below.
Im not sure if these are correct?
thanks
General Terms
cos x = ∑ (-1)n (x^(2n) / (2n)!)
COS x = ∑ (-1)n (x^(2n+1)/ (2n +1) / (2n)!)...
For my homework in my class, I'm supposed to find the first 3 non-zero terms of the Maclaurin series of f(x) = Integral sin(t)/t dt evaluated from 0 to x. I'm fairly sure that sint/t dt has no integral, so I'm lost in my search for the solution. Can you guys point me in the right direction?
MATLAB HELP! -- Maclaurin series
Hi, i have absolutely no programming experience with MATLAB and really need it. We have been assigned 2 problems using MATLAB and a bunch of others that don't need matlab. I was wondering if someone could show me what to do for the two needed to be done in...
Represent the function 5ln(7-x) as a power series, i.e., Maclaurin series,
C_0=
C_1=
C_2=
C_3=
C_4=
i got C_0 = 5 ln (7-0)
and i think C_1 = 5/(7-1)
but its wrong
the textbook says that C_1 will be the derivative of C_0
anyway... please give me some hint
Hi, hopefully you guys can point me in the right direction. The problem says to determine the number of terms necessary to approximate cos(x) to 8 significant digits using Maclaurin series approximation for x = .3*pi. I don't really have a problem with the actually question, my problem comes...
I have the Maclaurin series for cos (x), is their a way to find its radius of convergence from that?
ALSO
Is there a trick to find the shorter version of the power series for the Maclaurin series, I can never seem to find it so instead of the long series with each term but like E summation (the...
I have to find the first three non-zero terms in the Maclaurin series for the function sec(x).
I guess I have to use the known Maclaurin series for cos(x) and doing 1/cos(x) series with long division. I tried that but didn't get anywere close to the right answer. Could anyone please help me?
So I'm studying Taylor Series (I work ahead of my calc class so that when we cover topics I already know them and they are easier to study..) and tonight I found a formula for taylor series and maclaurin series, and i used them to prove eulers identity. However, I don't really know much about...
obtain the maclaurin series expansions of the following:
ln(1+e^x)
ok I am quite lost..i assume you set it equal to f(x) then differentiate..but what happens when you differentiate that?
also question (b) is
\frac{x}{(e^x-1)}
does that work down to
(e^x - 1)^-x ?
and if so..would you then...
I've looked at a number of explanations for the Maclaurin series of cos(x) yet none have given an easily understood answer, i was wondering if anyone has a way of explaining it when it is used as only a part of a function
eg. use that Maclaurin series of cos(x) to obtain the Macalurin series...
Can you please offer any hints or suggestions on how to do these two problems:
1) Find the Maclaurin series of (x^2 + 1)/(3x^2 + 2x - 1).
Should I perform long-division first? I can't seem to find any repeating pattern...
2) Evaluate the integral sqrt(12-4x-x^2) from x=2 to x=6.
I...
My question is as follows: Let f (x) = (1+x)^(1/2) – (1-x)^(1/2). Find the Maclaurin series for f(x) and use it to find f ^5 (0) and f ^20 (0).
I got: X + Riemann Sum { [ (-1)^(n-1) 1x3x5**x(2n-3) ] / (2^n) x n!} X^n (after combining two Riemann Sums together). And I got (7!5!) / 16 5! =...
I need some peer review of this quesion. Let f(x)=x^2e^{-2x^2}
(a) find the maclaurin series of f
(b) find the 100th degree maclaurin polynomial of f.
for part a i have:\sum_{n=0}^{\infty}\frac{(-2)^nx^{2n+2}}{n!}
and for b:\sum_{n=0}^{99}\frac{(-2)^nx^{2n+2}}{n!}
I'm supposed to find the following function as a Maclaurin Series. Please check if I'm correct.
f(x) = ln(1+x)
\sum \frac{\(x^n)((-1)^{n+1})}{n}
and that sum goes from n=1 to \infty
I also have to find the following functions as power series so please check it for me!
f(x) =...
i have trouble expandind f(x)= (sin2x)^2 into Maclaurin series
for sin(x) Maclaurin series is
\sum^{\infty}_{n=0} (-1)^n \frac{x^{2n+1}}{(2n+1)!}
probably the key is to change (sin2x)^2 into new shape. I found that
(sin2x)^2=2sin(2x^2), but that coefficent 2 is bothering me, what to do?
We learned that the Maclaurin series for sin(x) was
\sum^{\infty}_{n=0} (-1)^n \frac{x^{2n+1}}{(2n+1)!}
Is the Maclaurin series for sin(2x) the same, except with x replaced by 2x?