What is the coefficient for x^27 in the power series expansion of 1/(1+x^9)?

[greenteacup]

Homework Statement

The function
$$f(x)=\frac{1}{1+x^{9}}$$
can be expanded in a power series
$$\sum^{\infty}_{0} a_{n}x^{n}$$
with center c = 0.
Find the coefficient
$$a_{27}$$
of
$$x^{27}$$
in this power series.

2. The attempt at a solution

I can get to:

$$\sum^{\infty}_{0} (-1)^{n}(-x^{9})^{n}$$

which I think is right, but I'm not sure how to find $$a_{27}$$. We didn't talk about it in class.

 

[Dick]

You don't want (-1)^n and (-x^9)^n to both have a '-' in them do you? What are the first few terms in the series when you write them out? a_27 is the coefficient of x^27, which is the n=3 term in your series. What is it?

 

[greenteacup]

Ohhh, okay, I think I understand now. So the coefficient would just be $$(-1)^{3}=-1$$?

 

[Dick]

Ohhh, okay, I think I understand now. So the coefficient would just be $$(-1)^{3}=-1$$?

Right.