Taylor Series : How to determine coefficient

[cyt91]

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.

 

[HallsofIvy]

It is impossible to see your images, even by "right clicking".

 

[cyt91]

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.

 

[EsponV]

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.

 

[lanedance]

how about using tex?

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

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

 

[Tomer]

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...

 

[vela]

Fixed your post.

Homework Statement

Let $f(x) = \frac{1}{x^2+x+1}$. Suppose $$f(x)=\sum_{n=0}^\infty c_n x^n$$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.

 

[HallsofIvy]

Okay, so you have written "$c_nx^n$" as $x^{3m}- x^{3k- 1}$. To find the coefficient of $x^{18}$, 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 "$x^{3(6)}= 18$" and it has coefficient 1. There is no integer such that 3m- 1= 18 so that adds nothing. The coeffiicent of $x^{18}$ is 1.

Now do 19 and 20.

 

[cyt91]

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.

 

[SammyS]

[Mod Note: Replaced large images with hyperlinks, Hoot]

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 !

 

[jayman16]

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?

 

[vela]

If you can't see it, follow Tomer's advice back in post #6.

 

[jayman16]

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?

 

[vela]

Yes, you got it.