Holomorphic Functions and their Power Series representations We know that a function is said to be analytic if it has a convergent power series representation. Can we guarantee at all that this power series will converge uniformly or absolutely, even on compact subsets of $\mathbb{C}$? 
Indeed if the power series converges absolutely and uniformly, the power series will converge and so the function will be analytic; but is there a converse? 

If the power series at $z_0$ has radius of convergence $R$, then that series converges absolutely and uniformly on closed disks $\{z: |z - z_0| \le r\}$ for $r < R$.  In particular, if the function is entire (i.e. analytic on all of $\mathbb C$, then its Taylor series about any point converges absolutely and uniformly on every compact subset of $\mathbb C$.
