(a) Determine the radius of convergence of each of the following power series:

$$\sum_{n \geqslant 1} \frac{x^{n}}{n !}, \quad \sum_{n \geqslant 1} n ! x^{n}, \quad \sum_{n \geqslant 1}(n !)^{2} x^{n^{2}}$$

(b) State Taylor's theorem.

Show that

$$(1+x)^{1 / 2}=1+\sum_{n \geqslant 1} c_{n} x^{n}$$

for all $x \in(0,1)$, where

$$c_{n}=\frac{\frac{1}{2}\left(\frac{1}{2}-1\right) \ldots\left(\frac{1}{2}-n+1\right)}{n !}$$