f(x) = x^3/(x-2)^2
need to show the power series representation and the radius of convergence.
well, we can find the power series the same way we find Taylor series. but you might like this little shortcut.
i'll get you started. note that $\displaystyle \frac {x^3}{(x - 2)^2} = x^3 \cdot \frac 1{(x - 2)^2} = x^3 \cdot \frac d{dx} \frac 1{2 - x} = x^3 \cdot \frac d{dx} \frac 1{1 - {\color{red}(x - 1)}}$
Now recall that $\displaystyle \frac 1{1 - x} = \sum_{n = 0}^\infty x^n$ for $\displaystyle |x| < 1$ and that you can do term by term differentiation for power series.