Paper 1, Section II, D

Analysis I | Part IA, 2014

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

limk3ksin(x/3k)=x,\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 n12nsin(x/3n)\sum_{n \geqslant 1} 2^{n} \sin \left(x / 3^{n}\right) converges absolutely for all xRx \in \mathbb{R}.

(b) Let (an)nN\left(a_{n}\right)_{n \in \mathbb{N}} be a decreasing sequence of positive real numbers tending to zero. Show that for θR,θ\theta \in \mathbb{R}, \theta not a multiple of 2π2 \pi, the series

n1aneinθ\sum_{n \geqslant 1} a_{n} e^{i n \theta}


Hence, or otherwise, show that n1sin(nθ)n\sum_{n \geqslant 1} \frac{\sin (n \theta)}{n} converges for all θR\theta \in \mathbb{R}.

Typos? Please submit corrections to this page on GitHub.