Suppose radius of convergence of $\sum c_n z^n$ is 1. ($c_n, z \in \mathbb{C}$)
(i)Here, if $\{c_n\}$ is monotonically decreasing and $\lim c_n = 0$, then $\sum c_n z^n$ converges at every point on the circle $|z|$, except possibly at $z=1$. (Of course, $c_n$ is assumed to be non-negative real here)
(ii) $\{|c_n|\}$ is monotonically decreasing and $\lim c_n = 0$, then $\sum c_n z^n$ converges at every point on the circle $|z|$, except possibly at $z=1$. ($c_n \in \mathbb{R}$)
(iii) Same as (ii), but $c_n \in \mathbb{C}$
I know the statement (i) is true. However, are (ii) and (iii) false? I think (ii) is at least true. Please give me a proof if it true, otherwise give me a counterexample. Help!