Analysis I | Part IA, 2001

(i) If a0,a1,a_{0}, a_{1}, \ldots are complex numbers show that if, for some wC,w0w \in \mathbb{C}, w \neq 0, the set {anwn:n0}\left\{\left|a_{n} w^{n}\right|: n \geq 0\right\} is bounded and z<w|z|<|w|, then n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n} converges absolutely. Use this result to define the radius of convergence of the power series n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n}.

(ii) If an1/nR\left|a_{n}\right|^{1 / n} \rightarrow R as n(0<R<)n \rightarrow \infty(0<R<\infty) show that n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n} has radius of convergence equal to 1/R1 / R.

(iii) Give examples of power series with radii of convergence 1 such that (a) the series converges at all points of the circle of convergence, (b) diverges at all points of the circle of convergence, and (c) neither of these occurs.

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