Examining convergence of the series. Examine convergence of the series $$1-\frac{1}{2}+\frac{1\cdot3}{2\cdot4}-\frac{1\cdot3\cdot5}{2\cdot4\cdot6}+\cdots$$
Let's denote $a_n=\frac{1 \cdot 3 \cdot \cdots \cdot (2n-1)}{2 \cdot 4 \cdot \cdots \cdot (2n)}$ and consider:
$$\frac{a_n}{a_{n+1}}=1+\frac{1}{2n+1}$$
So by Gauss test we have absolute divergence and conditional convergence by Leibniz test and for limit 0
You have
$$a_n=(-1)^n\frac{\prod_{k=1}^n (2k-1) } {\prod_{k=1}^n (2k) }=\binom{-\frac{1}{2}}{n}$$ and then
$$S=\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty \binom{-\frac{1}{2}}{n} x^n=\frac{1}{\sqrt{1+x}}$$