Analysis I | Part IA, 2008

Let n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n} be a complex power series. Prove that there exists an R[0,]R \in[0, \infty] such that the series converges for every zz with z<R|z|<R and diverges for every zz with z>R|z|>R.

Find the value of RR for each of the following power series: (i) n=11n2zn\sum_{n=1}^{\infty} \frac{1}{n^{2}} z^{n}; (ii) n=0zn!\sum_{n=0}^{\infty} z^{n !}.

In each case, determine at which points on the circle z=R|z|=R the series converges.

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