This is a theorem in the Bak-Newman book of Complez analysis. It is stated:
Theorem 1: If $\sum_{n=0}^\infty a_nz^n$ has radius of convergence $R<\infty$ and, for every $n$, $a_n$ is real and $\ge 0$ then $f(z)=\sum_{n=0}^\infty a_nz^n$ has a singularity at $z=R$
I think I'm not understanding the statement correctly: the previous theorem is
Theorem 2: If $\sum_{n=0}^\infty a_nz^n$ has a positive radius of convergence $R$ then $f(z)=\sum_{n=0}^\infty a_nz^n$ has a singularity on the circle $|z|=R$
First of all, when talking about singularities at $z_0$, shouldn't we be working with an open (deleted) neigborhood of $z_0$? The way the theorem is stated doesn't make clear if we have a function defined on some open $D$ containing $\overline{D(0;R)}\setminus\{R\}$ and then the power series happens to have a singularity. I guess the question boils down to if a singularity (and what tyoe it is) can be determined by approaching it from only some directions.
Next, if there is a removable singularity, then the power series expansion "doesn't care" and cannot count as a singularity so the question is only about poles or essential singularities.
The power series $\sum \frac{z^n}{n^2}$ satisfies the conditions of he first theorem so there must be a singularity at $z=R=1$, where $\sum \frac{1}{n^2}<\infty$. Does it mean that the singularity is essential?
Can someone clarify all of this? Thanks!
The concept of singularity that the authors are using here is taken from
Definition 18.1: Suppose that $f$ is analytic in a disc $D$ and that $z_0\in\partial D$. Then $f$ is said to be regular at $z_0$ if $f$ can be continued analytically to a region $D_1$ with $z_0\in D_1$. Otherwise, $f$ is said to have a singularity at $z_0$.
So, this has nothing to do with removable or essential singularities, which are indeed concepts related to analytic functions defined on a deleted neighborhood of a point.