Question

find a power series centered at x=0 for the functions and define their domain
f(x)=1/3-x

Answer 1

Recall the power series for \(f(x)=\dfrac{1}{1-x}\):
\[f(x)=\sum_{n=0}^\infty x^n\text{ for }|x|<1\]
For the given function, in order to make it look more accommodating, you just have to do some minor algebraic manipulation:
\[\frac{1}{3-x}=\frac{1}{3}\cdot\frac{1}{1-\dfrac{x}{3}}=\frac{1}{3}\sum_{n=0}^\infty \left(\frac{x}{3}\right)^n\]
which converges for \(\left|\dfrac{x}{3}\right|<1\), or \(|x|<3\).