# Paper 1, Section II, F

(a) Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series with $a_{n} \in \mathbb{C}$. Show that there exists $R \in[0, \infty]$ (called the radius of convergence) such that the series is absolutely convergent when $|z| but is divergent when $|z|>R$.

Suppose that the radius of convergence of the series $\sum_{n=0}^{\infty} a_{n} z^{n}$ is $R=2$. For a fixed positive integer $k$, find the radii of convergence of the following series. [You may assume that $\lim _{n \rightarrow \infty}\left|a_{n}\right|^{1 / n}$ exists.] (i) $\sum_{n=0}^{\infty} a_{n}^{k} z^{n}$. (ii) $\sum_{n=0}^{\infty} a_{n} z^{k n}$. (iii) $\sum_{n=0}^{\infty} a_{n} z^{n^{2}}$.

(b) Suppose that there exist values of $z$ for which $\sum_{n=0}^{\infty} b_{n} e^{n z}$ converges and values for which it diverges. Show that there exists a real number $S$ such that $\sum_{n=0}^{\infty} b_{n} e^{n z}$ diverges whenever $\operatorname{Re}(z)>S$ and converges whenever $\operatorname{Re}(z).

Determine the set of values of $z$ for which

$\sum_{n=0}^{\infty} \frac{2^{n} e^{i n z}}{(n+1)^{2}}$

converges.