Analysis | Part IA, 2005

Let n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n} be a power series in the complex plane with radius of convergence RR. Show that anzn\left|a_{n} z^{n}\right| is unbounded in nn for any zz with z>R|z|>R. State clearly any results on absolute convergence that are used.

For every R[0,]R \in[0, \infty], show that there exists a power series n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n} with radius of convergence RR.

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