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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!

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@thomas $c_n$ might be an alternating series in (ii) –  Katlus Aug 25 '12 at 9:44

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up vote 3 down vote accepted

That (ii) is false is implied by the exception in (i). For any series according to (i) that doesn't converge at $z=1$, e.g. $c_n=1/n$, change the signs of the odd coefficients to obtain a series according to (ii) that doesn't converge at $z=-1$.

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+1. Likewise, for every $z_0$ such that $|z_0|=1$, multiplying $c_n$ by $z_0^{-n}$ moves the point where one knows that the series diverges to $z_0$. –  Did Aug 25 '12 at 10:44

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