Question: Examine convergence of the series $$1-\frac{1}{2}+\frac{1\cdot3}{2\cdot4}-\frac{1\cdot3\cdot5}{2\cdot4\cdot6}+\cdots$$

Answer: 
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](https://math.stackexchange.com/questions/1899857/mathematical-problem-induction-frac12-cdot-frac34-cdots-frac2n-12n-frac)
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
$$S=\sum\_{n=0}^\infty a\_n x^n=\sum\_{n=0}^\infty \binom{-\frac{1}{2}}{n} x^n=\frac{1}{\sqrt{1+x}}$$