Exercise 2, § 3

Let $(z_n)$ be an infinite sequence of complex numbers $z_n = x_n + i y_n$ such that $x_n \geq 0$ for all $n$. Show that if the series whose general terms are $z_n$ and $z_n^2$ are convergent, then the series whose general term is $z_n^2$ is absolutely convergent. Give an example in which this result is false when the condition $x_n \geq 0$ (which can also be written $-\pi/2 \leq \operatorname{Am}(z_n) \leq \pi/2$) is replaced by

no matter how small the real number $\varepsilon > 0$ may be. (Here $\operatorname{Am}(z)$ denotes that radian measure of the amplitude of $z_n$ which belongs to the interval $]-\pi, +\pi]$.)