For each of the following series, determine for which real numbers $x$ it diverges, for which it converges, and for which it converges absolutely. Justify your answers briefly.

(a) $\sum_{n \geqslant 1} \frac{3+(\sin x)^{n}}{n}(\sin x)^{n}$,

(b) $\quad \sum_{n \geqslant 1}|\sin x|^{n} \frac{(-1)^{n}}{\sqrt{n}}$,

(c)

$$\sum_{n \geqslant 1} \underbrace{\sin (0.99 \sin (0.99 \ldots \sin (0.99 x) \ldots))}_{n \text { times }} .$$