Solution to Laplace's equation on the disk I am reading through this document http://web.math.ucsb.edu/~grigoryan/124B/lecs/lec10.pdf, which arrives at the following series solution to Laplace's equation on the disk:
$$u(r, \theta)=\frac{A_{0}}{2}+\sum_{n=1}^{\infty} r^{n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right).$$


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*Why do we sum up all the eigenfunctions? How do we know this gives a solution? (I know very little about diff. equations, and I'm just looking for a brief explanation here.)

*What is the nature of the convergence of this series? Can we pick $A_n$ and $B_n$ that decay rapidly enough so that  it converge uniformly on every compact subset? 

This is just a formal solution. If you just differentiate termwise ignoring any convergence issues this should come out as an eigenfunction. The tricky part is now to find which coefficients to pick for $A_n$ and $B_n$ to satisfy the boundary conditions and then prove some kind of convergence.
