Does a power series always have a singularity on its radius of convergence?

Question

Suppose $P(z)$ is a power series with radius of convergence $R>0$. Must there be some $c$ with $|c| = R$ such that the limit of $P(z)$ as you approach $c$ from within the open disc of radius $R$ is infinity?

Answer 1

A singularity -- yes.

A point such that $\lim_{z\to c}P(z)=\infty$ -- no. Consider $f(z)=e^{1/(z-1)}$, which has an essential singularity at $z=1$. This implies that "anything" can happen as $z\to 1$.

Answer 2

Almost anything can happen at the boundary. For example, $$ f(z) = \sum_{n=1}^\infty \frac{z^n}{n^2} $$ is bounded on the unit disc, but still can't be extended analytically to any disc of radius greater than $1$.

In general, it doesn't make sense to speak of poles, or essential singularities at the boundary of the disc of convergence. Those terms are only used for isolated singularities, and for "most" power series with a finite radius of convergence, the corresponding analytic function can't be extended across any boundary point.