I am working on the power series.
Here is the question
$$f(x)=\frac {9}{1+100x^2}$$
represented as a power series
$$f(x) = \sum^{\infty}_{n=0}c_nx^n$$
I need to find $c_0,c_1,c_2,c_3,c_4,R$
I got this
$c_0=9$
$c_1=-90$
$c_2=1800$
$c_3=-54000$
$c_4=2160000$
$R= \frac {1}{10}$
I know that $c_{1-4}$ are wrong. I don't know why
I got the summation to be
$$\sum^{\infty}_{n=0}9(-10x)^n$$
$$9-90x+900x^2-9000x^3+90000x^4$$
taking derivatives to find the $c_n$
Best Answer
HINT: Rewrite $f(x)$ as
$$\dfrac{9}{1 + 100x^2} = 9 \cdot \dfrac{1}{1 - (-100x^2)}$$
Use the following identity to write $f(x)$ as the power series:
$$\dfrac{1}{1 - g(x)} = 1 + g(x) + (g(x))^2 + (g(x))^3 + \cdots = \sum\limits_{n = 0}^{\infty} (g(x))^n$$
Now answer the problem, using the info above. It is easier to write out partial sum of the series consisting of $5$ terms.