I found an unclear step for me in Zorich, Mathematical Analysis I, sec. 5.5, pag. 270. We are trying to find all $z \in\mathbb{C}$ for which the series:
$$c_0+c_1(z-z_0)+c_2(z-z_0)^2+...$$
converges. To do this, we try to understand when it converges absolutely, applying the Cauchy criterion to the series:
$$|c_0|+|c_1(z-z_0)|+|c_2(z-z_0)^2|+...$$
obtaining that it converges if:
$$|z-z_0|< \frac{1}{\overline{\lim}_{n\to\infty}\sqrt[n]{|c_n|}} $$
and all is ok. Now, Zorich says (referring to the series with absolute values): the general term does not tend to zero if $|z-z_0| \geq \frac{1}{\overline{\lim}_{n\to\infty}\sqrt[n]{|c_n|}} $.
I can't completely understand this sentence, in particular the reason why he also includes the case $|z-z_0| =\frac{1}{\overline{\lim}_{n\to\infty}\sqrt[n]{|c_n|}} $. In this case I would have:
$$|c_0|+\frac{|c_1|}{\left (\overline{\lim}_{n\to\infty}\sqrt[n]{|c_n|} \right )} +\frac{|c_2|}{\left( \overline{\lim}_{n\to\infty}\sqrt[n]{|c_n|}\right )^2} +...$$
and so I would verify that:
$$\lim_{k\to\infty}\frac{|c_k|}{\left( \overline{\lim}_{n\to\infty}\sqrt[n]{|c_n|}\right )^k}\neq 0$$
but I can't. How does Zorich manage to say that for all $z\in\mathbb{C}$, such that $|z-z_0| =\frac{1}{\overline{\lim}_{n\to\infty}\sqrt[n]{|c_n|}} $, the series with absolute values diverges?
Thanks.