Taylor Series : How to determine coefficient

Another way to think about it is that you are essentially comparing the coefficient of each power of x on the left side with the coefficient on the right side. For example, the coefficient of x^18 on the left is c_18, and the coefficient on the right is 1. Since these two must be equal, c_18 must be equal to 1. Similarly, the coefficient of x^19 on the left is c_19, and the coefficient on the right is -1. Therefore, c_19 must be equal to -1. And since there is no x^20 term on the right side, there must be no x^20 term on the left side, meaning c_20 must be equal to 0
  • #1
cyt91
53
0

Homework Statement


https://skydrive.live.com/?cid=6b041751c72e14ad#!/?cid=6b041751c72e14ad&sc=photos&uc=3&id=6B041751C72E14AD%21149!cid=6B041751C72E14AD&id=6B041751C72E14AD%21154&sc=photos


The Attempt at a Solution



https://skydrive.live.com/?cid=6b041751c72e14ad#!/?cid=6b041751c72e14ad&sc=photos&uc=3&id=6B041751C72E14AD%21149!cid=6B041751C72E14AD&id=6B041751C72E14AD%21155&sc=photos

https://skydrive.live.com/?cid=6b041751c72e14ad#!/?cid=6b041751c72e14ad&sc=photos&uc=3&id=6B041751C72E14AD%21149!cid=6B041751C72E14AD&id=6B041751C72E14AD%21156&sc=photos


Please view the images posted(right-click,view image if not loaded properly) for the question and my attempts at the solution.
How do we determine the coefficient of c18,c19 and c20 respectively?
Thank you for your help.
 
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  • #2
It is impossible to see your images, even by "right clicking".
 
  • #3
I've tried loading the pictures(by right-clicking and selecting open in new tab). They loaded just fine.

Sorry,I'm not that good in Latex.
 
  • #4
I think the issue is that we can't find the link to click on in the first place. Nothing in your original statement contains a hyperlink - it's all just plaintext. Unless I'm starting to go blind.
 
  • #5
how about using tex?

right click on code below to see how its written
[tex]
f(x) = \Sum_n\frac{1}{n!}\frac{d^{(n)}f(0)}}{dx}x^n[/tex]
[/tex]

you'll generally get a quicker answer, and its easier for people to cut/paste&edit your original post
 
  • #6
Hi cyt91,

well, I believe you did most of the job. Try to write the actual sum you just found, element by element... You can easily see what the coefficients of x18, x19 and x20 are...
 
  • #7
Fixed your post.
cyt91 said:

Homework Statement



Let [itex]f(x) = \frac{1}{x^2+x+1}[/itex]. Suppose [tex]f(x)=\sum_{n=0}^\infty c_n x^n[/tex]is the Taylor series representation for f(x) at x=0. Find the value of c18-c19+c20.


The Attempt at a Solution



\begin{align*}
f(x) &= \frac{(1-x)}{(1-x)(x^2+x+1)} = \frac{1-x}{(x^2+x+1)+(-x^3-x^2-x)} = \frac{1-x}{1-x^3} \\
&= \frac{1}{1-x^3} - x\left(\frac{1}{1-x^3}\right) \\
&= \sum_{n=0}^\infty x^{3n} - x\sum_{n=0}^\infty x^{3n} \\
&= \sum_{n=0}^\infty x^{3n} - \sum_{n=0}^\infty x^{3n+1}
\end{align*}

Please view the images posted(right-click,view image if not loaded properly) for the question and my attempts at the solution.
How do we determine the coefficient of c18,c19 and c20 respectively?
Thank you for your help.
 
  • #8
Okay, so you have written "[itex]c_nx^n[/itex]" as [itex]x^{3m}- x^{3k- 1}[/itex]. To find the coefficient of [itex]x^{18}[/itex], say, you need to find m so that 3m= 18 and k such that 3k- 1= 18. If there is no such integer, the coefficient is 0. 3m= 18 gives m= 6 so there is a term "[itex]x^{3(6)}= 18[/itex]" and it has coefficient 1. There is no integer such that 3m- 1= 18 so that adds nothing. The coeffiicent of [itex]x^{18}[/itex] is 1.

Now do 19 and 20.
 
  • #9
Hi.

Thanks a lot for your help.
I've found c_19 = -1 and c_20 = 0.

Just one more clarification: when we are finding c_19 / c_18 /c _20 ,
we are in fact finding the 19th / 18th/ 20th order of Taylor's polynomial right?

So,for instance,if we want to look for c_18, we'll be looking at 3n = 18 (since n is an integer)
i.e. the 18th order of Taylor's polynomial.
 
  • #10
[Mod Note: Replaced large images with hyperlinks, Hoot]
cyt91 said:

Homework Statement


https://byfiles.storage.live.com/y1p2z6eR8OToFBvNcV05JbbM1tfydTwF_jlCbyIBdnrB7hdMAp99X_itLOr3QdNX28aHJE-sB5AzP8/Untitled.png?psid=1


The Attempt at a Solution



https://byfiles.storage.live.com/y1pB-lBs8QHND8-V293IdHCDOwnJGmyNO_Tb2R-Lq8BfKGElu1jZP-0aUQMdIXbXa1gGbf1PwKHKtk/04092011158.jpg?psid=1

https://byfiles.storage.live.com/y1pzWQO-7xvKIfg3makuohkivYAG80FIioxSFBrV1WKAnVpG3QsxkyPiTlelEPExSw8pB-iIGh_k8Y/04092011159.jpg?psid=1


Please view the images posted(right-click,view image if not loaded properly) for the question and my attempts at the solution.
How do we determine the coefficient of c18,c19 and c20 respectively?
Thank you for your help.
This is the Original Post with the corrected Image links.

Hope that helps someone !
 
Last edited by a moderator:
  • #11
still don't understand how the coefficients c_18, 19, and 20 are obtained. can you put it in even simpler terms why c_18 is 1, 19 is -1 and 20 is 0?
 
  • #12
If you can't see it, follow Tomer's advice back in post #6.
 
  • #13
i got it. thanks. would i be right in saying that the the only term with x^18 in the series would be coming from summation 0 to infinity (x^3n) when n = 6 on the right hand side. c_18x^18 = x^18 meaning c_18 = 1. Similarly the only term with x^19 on the right hand side is when n = 6 where it is -(x^3n+1) = - (x^19). Hence c_19x^19 = -(x^19) , c_19 is therefore -1 and since there are no integer values for n that can give me x^20 on the right hand side i can conclude that there is no x^20 term on the right. So c_20 must be 0 on the left removing the x^20 term on the left. Is this the reasoning?
 
  • #14
Yes, you got it.
 

FAQ: Taylor Series : How to determine coefficient

What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function with a polynomial, making it easier to work with and calculate. The series is named after the mathematician Brook Taylor.

How do you determine the coefficients in a Taylor series?

The coefficients in a Taylor series can be determined using the Taylor series formula, which involves taking the derivatives of the function at a specific point and plugging them into the formula. Alternatively, you can also use the Maclaurin series, which is a special case of the Taylor series where the point of expansion is 0.

What is the purpose of determining coefficients in a Taylor series?

The coefficients in a Taylor series help us to approximate a function with a polynomial, making it easier to work with and calculate. This allows us to solve complex problems and equations more efficiently and accurately. It also helps us to better understand the behavior of a function near a specific point.

Can you use Taylor series to approximate any function?

Yes, Taylor series can be used to approximate any function as long as the function is infinitely differentiable at the point of expansion. However, the accuracy of the approximation may vary depending on the function and the number of terms used in the series.

How do you know when to stop adding terms in a Taylor series?

The accuracy of a Taylor series approximation depends on the number of terms used in the series. Generally, the more terms you add, the more accurate the approximation will be. However, in practice, it is often enough to use a few terms to get a good approximation. To determine when to stop adding terms, you can use a remainder term formula to estimate the error of the approximation and choose a desired level of accuracy.

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