Exercise 8, § 4

a) Let $(z_n)$ be a sequence of complex numbers. Show that if the series with general term $z_n, z_n^2, \ldots, z_n^{q-1}, |z_n|^q$ converge, then the infinite product with general factor $1 + z_n$ converges.

b) If $z_n$ is real and if the series with general term $z_n$ converges, then the infinite product with general factor $1 + z_n$ converges if the series with general term $z_n^2$ converges, and one has

c) For every integer $p \geqslant 2$ put

Define the sequence $(z_n)$ of complex numbers in the following way: for $n = h_p + m$, $0 \leq m \leq k_p - 1$, put $z_n = e^{m i \omega_p} / \log p$. Show that for every integer $q > 0$ the series with general term $z_n^q$ converges, but that the product with general factor $1 + z_n$ does not converge.