Let $P(z)$ be of the form $\sum_{n=0}^{\infty}a_nz^n$.
It is known that
Lemma
$(i)$if $P(z_o)$ converges then all $P(z)$ with $|z|<|z_o|\in\mathbb{C}$ absolutely converge.
$(ii)$ if $P(z_0)$ converges absolutely then all $P(z)$ with $|z|\leq|z_o|\in\mathbb{C}$ absolutely converge
Define $R:=sup\{|z|:P(z)$ converges$\}$ and
$R':=sup\{|z|:P(z)$ converges absolutely$\}$
With the lemma one can conclude that $R=R'$
One also has the notation $D_R(0)=\{z\in\mathbb{C}:z<R\}$ and $\bar{D}_R(0)=\{z\in\mathbb{C}:z\leq R\}$
Now my questions:
Can somebody give me an example where $\{z\in\mathbb{C},|z|=R\}$ but it is not clear whether the elements converge or converge absolutely?
Second question why does the convergencebehaviour solely depends on the coefficients i.e. $(a_n)_{n\in\mathbb{N}_0}$
First Question:
In general, as you noted, the power series on $\{z\in\mathbb{C},|z|=R\}$ could converge for some values of $z$ and diverge for others. An example would be:
$\sum_{n=1}^{\infty }\frac{z^n}{n}$ which converges for all $|z|=1, z\neq 1$ and diverges for $z=1$.
Second question:
The Lemmas you've stated tell you that once you have a point $z$ at which there is convergence, you automatically have convergence for all those points with smaller modulus. The Cauchy-Hadamard theorem gives you the dependence of the supremum of the moduli of those $z$ for which you have convergence with respect to your sequence $(a_n)_{n\in\mathbb{N}_0}$. So then, strictly within your radius of convergence you have convergence and strictly outside you have divergence. But exactly at your radius of convergence, the behavior does depend on $z$.