complex analysis – find Taylor's series but the center is not analytic

I want to find a Taylor series for the complex function

$$ f (z) = dfrac {z ^ 2} {2 + z}, $$ focused on $ z = $ 2.

I've found the taylor series $ f (z) $ focused on $ z = $ z_0,
$$ f (z) = (z-2) + dfrac {4} {z + 2} = (z-2) +4 sum limits_ {n = 0} ^ infty dfrac {(- 1) ^ n} {(2 + z_0) ^ {n + 1}} (z-z_0) n. $$

Since $ f (z) $ is not analytic to $ z = $ 2, can we find Taylor's series?