example of a convergent series that $\lim \sup |\frac{z_{n+1}}{z_n} | > 1$

Question

Let $(z_n) \subset \mathbb C$, with $z_n \neq 0$. It's known that if $\lim \sup |\frac{z_{n+1}}{z_n} | < 1$, so $\sum |z_n|$ converges, then $\sum z_n$ converges.

Can I find an example of a convergent series that $\lim \sup |\frac{z_{n+1}}{z_n} | > 1$ ?

Answer 1

What about the sequence \begin{align*} z_n&=\begin{cases} 5 \left(\frac12\right)^{n-1}&n\text{ odd}\\ \left(\frac12\right)^n&n\text{ even} \end{cases} \end{align*}

Answer 2

Let the terms be $$1,2,\frac{1}{2},1,\frac{1}{4},\frac{1}{2},\frac{1}{8},\frac{1}{4},\frac{1}{16},\frac{1}{8}, \frac{1}{32},\dots.$$