(a) Show that for all $x \in \mathbb{R}$,

$$\lim _{k \rightarrow \infty} 3^{k} \sin \left(x / 3^{k}\right)=x,$$

stating carefully what properties of sin you are using.

Show that the series $\sum_{n \geqslant 1} 2^{n} \sin \left(x / 3^{n}\right)$ converges absolutely for all $x \in \mathbb{R}$.

(b) Let $\left(a_{n}\right)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}, \theta$ not a multiple of $2 \pi$, the series

$$\sum_{n \geqslant 1} a_{n} e^{i n \theta}$$

converges.

Hence, or otherwise, show that $\sum_{n \geqslant 1} \frac{\sin (n \theta)}{n}$ converges for all $\theta \in \mathbb{R}$.