# Thread: Find the power series of this function

1. ## Find the power series of this function

f(x) = x^3/(x-2)^2

need to show the power series representation and the radius of convergence.

2. Originally Posted by rcmango
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 $\frac {x^3}{(x - 2)^2} = x^3 \cdot \frac 1{(x - 2)^2} = x^3 \cdot \frac d{dx} \frac 1{2 - x} = x^3 \cdot \frac d{dx} \frac 1{1 - {\color{red}(x - 1)}}$

Now recall that $\frac 1{1 - x} = \sum_{n = 0}^\infty x^n$ for $|x| < 1$ and that you can do term by term differentiation for power series.