Suppose $\sum a_nz^n$ has a radius of convergence R and let C be the circle {$z \in \mathbb{C}: |z| = R$} Prove or disprove the following:
a) If $\sum a_nz^n$ converges at some point on C then it converges everywhere on C
b) If $\sum a_nz^n$ converges absolutely at some point on C then it converges absolutely everywhere on C
c) If $\sum a_nz^n$ converges at every point on C, except possibly one, then it converges absolutely everywhere on C (hint: the series $\sum z^n/n$)
I believe that a) and b) are true. For a) I have
There exists $M\geq 0$ such that $|a_nz_0^n| \leq M$, then if $|z|<|z_0|$, $|a_nz^n| = |a_nz_0|^n|\frac{z}{z_0}|^n \leq M |\frac{z}{z_0}|^n$. Comparing this to a convergent geometric we have it is convergent everywhere.
For b) it seems like it's the same proof as part a) and c) I am lost.