About complex sequences and series

Question

I have to find a complex sequence $z_{n}$ such that $\sum\limits_{n\geq 1}z_{n}$ and $\sum\limits_{n\geq 1}z_{n}^{2}$ are convergent series but $\sum\limits_{n\geq 1}|z_{n}|^{2}$ is not a convergent serie.

I think that it is possible if we use a complex sequence such that $\text{Re}(z_{n})\ngeqslant 0$ $\forall n\in\mathbb{N}$ but I don't know how to choose that $z_{n}$ explaining in detail why $\sum\limits_{n\geq 1}|z_{n}|^{2}$ is not convergent.

Answer 1

Take $z_n=\dfrac{i^n}{\sqrt n}$. Then$$(\forall n\in\mathbb N):\lvert z_n\rvert=\frac1n$$and the harmonic series diverges. But both series$$\sum_{n=1}^\infty z_n\text{ and }\sum_{n=1}^\infty{z_n}^2$$converge, by Dirichlet's test.

Answer 2

Take $z_n=\frac{\sin{n}}{\sqrt{n}}+i\frac{\sin{n}}{\sqrt{n}}$