TITLE: Does a bounded convergent power series on an open disc extend to the boundary?
QUESTION [6 upvotes]: Here is my question: Suppose that $|\sum_{n=0}^{\infty}a_nz^n| \leq M$ for all $z \in D_r$ (the open disc or radius $r$). Does this power series converge on $\partial D_r$?

REPLY [3 votes]: The answer is no. There are even functions that are continuous on $\bar D_1$ and analytic  on $D_1$ whose Taylor series diverges for some points on the boundary. See this question