Paper 1, Section I, D

Analysis I | Part IA, 2010

Let n0anzn\sum_{n \geqslant 0} a_{n} z^{n} be a complex power series. State carefully what it means for the power series to have radius of convergence RR, with R[0,]R \in[0, \infty].

Suppose the power series has radius of convergence RR, with 0<R<0<R<\infty. Show that the sequence anzn\left|a_{n} z^{n}\right| is unbounded if z>R|z|>R.

Find the radius of convergence of n1zn/n3\sum_{n \geqslant 1} z^{n} / n^{3}.

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