Suppose that $∑a_nz^n$ has radius of convergence $R$ and let $C$ be the circle {$z ∈ C : |z| = R$}. Prove or disprove the following (which may or may not be true):
(i) If $∑ a_nz^ n$ converges at some point on $C$ then it converges everywhere on $C$.
(ii) If $∑ a_nz^ n$ converges absolutely at some point on $C$ then it converges absolutely everywhere on C.
(iii) If $∑ a_nz^ n$ converges at every point on $C$, except possibly one, then it converges absolutely everywhere on $C$.
Sorry since this question has already been asked, but I am thinking of solving it in the following way, for
i) it's false since we can put $z=1$ and it will diverge, and $z=-1$ and it will converge.
ii) I think it's true since the points on the radius all have the same absolute value, thus the absolute value of their series converge, thus they converge.
iii) I am not sure how to go with it
i) You should provide an example, such as, for instance, $\sum_{n=1}^\infty\frac{z^n}n$, which converges when $z=-1$ and diverges when $z=1$.
ii) You are right.
iii) Again, take $\sum_{n=1}^\infty\frac{z^n}n$, which converges when $|z|=1$ and $z\ne1$ (by the Dirichlet test) but diverges when $z=1$. This shows that the assertion is false.