Does the following series represent a harmonic function in an annulus?

Question

It is known that given an annulus $\Delta=\Delta(0;R_1,R_2)\subset\mathbb{C}$ and a harmonic function $u:\Delta\to\mathbb{R}$, then $u$ admits in $\Delta$ of a representation

$$u(re^{i\theta})=k\log r+a_0+\sum_{n\in\mathbb{Z}\setminus\{0\}}(a_n\cos n \theta+b_n\sin n \theta)r^n \tag{1}$$

where the series converges absolutely and almost uniformly in $\Delta$. Conversly, I was wondering whether a function of the form (1), assuming that the convergence is absolute and almost uniform in $\Delta$, represents a harmonic function in $\Delta$ or there is a counterexample.

Answer 1

Yes. Each term is harmonic (easy to check, just differentiate), so partial sums are harmonic, and uniform limit of harmonic functions is harmonic. By almost uniform convergence I assume you mean convergence on compact subsets, which is the correct convergence you obtain here.

By the way another way to see this is to rewrite this function as a real part of a holomorphic function plus $k \log |z|$.