Coefficients of power series with binomial theorem I am trying to get the coefficients of the power series of $\sqrt{\frac{1+z}{1-z}}$.
Rewriting gets us to  $(1 + \frac{2z}{1-z})^{0.5}$. Now using the binomial theorem gets us $\sum_{k=0}^n {\frac{1}{2}\choose k} (\frac{2z}{1-z})^k $ but didnt get far with that.

You could write $$\sqrt{\frac{1+x}{1-x}}=\frac{1+x}{\sqrt{1-x^2}}$$ and then expand $$(1+x)(1-x^2)^{-\frac12}$$ and get $$1+x+\frac12x^2+\frac12x^3+\frac38x^4+\frac38x^5+\frac{5}{16}x^6+\frac{5}{16}x^7+...$$
This is convenient since all the coefficients occur in identical pairs...
