Two Taylor expansions of $\frac{1}{1+\sqrt{2-z}}$ about $z=0$ 
How do you start expanding this function $$f(z)= \frac{1}{1+\sqrt{2-z}}$$ into two Taylor expansions about $z=0$? 

The best I came up is to let $u=\sqrt{2-z}$ and then expand $f(z)$ as a geometric series.

Alternatively, with some algebraic manipulations you get at:
$$
  f(z) = \frac{1}{1+\sqrt{2-z}} = \frac{1-\sqrt{2-z}}{\left(1+\sqrt{2-z}\right)\left(1-\sqrt{2-z}\right)} = \frac{1-\sqrt{2-z}}{z-1} = \frac{\sqrt{2-z}-1}{1-z}
$$
Suppose, you worked out $\sqrt{2-z}-1 = \sum_{n=0}^\infty c_n z^n$, then
$$
  \frac{\sqrt{2-z}-1}{1-z} = \sum_{n=0}^\infty \left( \sum_{m=0}^n c_m \right) z^n
$$
