I've been given a function $f(x)=\dfrac{2}{1+9x^2}$ and been asked to get the first few coefficients of its power series representation.
This looks like the series $\sum\limits_{n=0}^{\infty}{a_n x^n}=\dfrac{a}{1-x}: |x|<1$, so I did the following to get the power series representation:
$$f(x)=\frac{2}{1+9x^2} \textrm{, so my } a_n=2 \textrm{ and my } r=-9x^2$$
$$f(x)=\sum_{n=0}^{\infty}{(2)(-9x^2)^n} \textrm{, if } |-9x^2|<1$$
So from what I understand, to get the first few coefficients I should solve $(2)(-9x^2)^n$ for $n=0,1,2,3,4$. Doing this, I got:
$$(2)(-9x^2)^0=2x^0$$
$$(2)(-9x^2)^1=-18x^2$$
$$(2)(-9x^2)^2=162x^4$$
$$(2)(-9x^2)^3=-1458x^6$$
$$(2)(-9x^2)^4=13122x^8$$
So my coefficients should be $2, -18, 162, -1458,$ and $13122$. However, only $2$ is correct and the rest are wrong.
What is wrong with my approach?
Best Answer
Your calculations are correct but you have to list coefficients of odd powers also. So the coefficients are $2,0,-18,0,162,0,...$.