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.

In reply to your comment on where you are blocked, consider that
$$
\begin{array}{l}
 \left( {\sum\limits_{0 \le n} {z^n } } \right)^k  = \left( {\frac{1}{{1 - z}}} \right)^k  = \left( {1 - z} \right)^{ - k}  =  \\ 
  = \sum\limits_{0 \le j} {\left( { - 1} \right)^j \left( \begin{array}{c}
  - k \\ 
 j \\ 
 \end{array} \right)z^j }  = \sum\limits_{0 \le j} {\left( \begin{array}{c}
 j + k - 1 \\ 
 j \\ 
 \end{array} \right)z^j }  \\ 
 \end{array}
$$
