I have to show that $$\frac{2}{1+(1+x)^{1/2}} = 1 - \frac{k}1\cdot\frac{x}4 + \frac{k(k+3)}{2!}\cdot\left(\frac{x}4\right)^2-\frac{k(k+4)(k+5)}{3!}\cdot\left(\frac{x}4\right)^3+\ldots$$
I need to start with $z = a + \frac{x}z$ to find the Lagrange expansion of $\frac{1}{z^k}$, then take $a=2$. I might be over thinking, but I do not know what it means to find the Lagrange expansion of $\frac{1}{z^k}$. I know what the Langrange expansion is, but I thought $z=a + \frac{x}{z}$ would mean we would find the Lagrange expansion of $\frac{1}z$.