Exercise 8, § 4

(a) Let $(z_n)$ be a sequence of complex numbers. Now prove that if the series of general term $z_n, z_n^2, \ldots, z_n^{q-1}, |z_n|^q$ are convergent, the infinite product of general factor $1 + z_n$ is convergent. (b) If $z_n$ is real and if the series of general term $z_n$ is convergent, the infinite product of general factor $1 + z_n$ is convergent if the series of general term $z_n^2$ is convergent, and one has

(c) For every integer $p$, one sets The sequence $(z_n)$ of complex numbers is defined as follows: for $n = h_p + m,\ 0 \leq m \leq k_p - 1$, one sets $z_n = e^{m i \omega_p}/\log p$. Now prove that for every integer $q > 0$, the series of general term $z_n^q$ is convergent, but that the product of general factor $1 + z_n$ is not convergent.