Find a power series that is convergent on the closed unit disk but diverges elsewhere.

Question: Does there exist a power series centered at $$z=0$$, $$f(z)=\sum_{n=0}^\infty a_n z^n$$ such that the domain of $$f$$ is exactly the unit disk $$D^2\subset \mathbb{C}$$? In other words, I'm looking for a power series whose radius of convergence $$\rho=1$$ such that the series also converges on the unit circle.

Motivation: I'm thinking about a problem: "does there exist a Laurent series that converges only on the unit circle but nowhere else?" I realize that this problem reduces to the above question.

We know $$f(z)$$ will converge if it converges absolutely (on $$D^2$$), i.e. $$\sum_{n=1}^\infty |a_n| \: |z|^n,$$ converges. Take $$a_n = 1/n^2$$. For $$|z| \leq 1$$ (i.e. $$z \in D^2$$), we have: $$\sum^\infty_{n=1} \frac{1}{n^2} |z|^n \leq \sum^\infty_{n=1} \frac{1}{n^2},$$ and the RHS converges by the $$p$$-test. Thus the LHS converges (since all terms are non-negative), implying absolute convergence of $$\sum_n \frac{1}{n^2} z^n$$. For $$|z| > 1$$, we see: $$\lim_{n \to \infty} \frac{1}{n^2} |z|^n \neq 0,$$ so that $$\sum_n \frac{1}{n^2} z^n$$ diverges on $$|z| > 1$$ by the divergence test for complex series.
Thus, $$f(z) = \sum_{n=1}^\infty \frac{1}{n^2} z^n$$ is an example of a function that meets your criteria. (Notice that I started at $$n= 1$$, but if you want to start at $$n= 0$$ you can take $$a_0$$ to be anything, say 1, and the argument still holds).
$$f(z)=\sum \frac{z^{2^n}}{2^n}$$ will do; it is obviously absolutely convergent on the closed unit disc, but if $$f$$ would be analytic at a point $$\alpha$$ on the unit circle (which means that there is an open set $$\alpha \in U$$ containing that on which $$f$$ extends analytically), then $$f'$$ would be too and $$f'(z)=\sum z^{2^n}$$ which is the well-known example of a function with natural boundary the circle and with the easy proof that $$f'(r\zeta) \to \infty, r \to 1$$ for any $$\zeta$$ root of unity of order some power of two of unit and those are dense in the unit circle.