Paper 1, Section I, 4E4 \mathrm{E}

Analysis I | Part IA, 2017

Show that if the power series n=0anzn(zC)\sum_{n=0}^{\infty} a_{n} z^{n}(z \in \mathbb{C}) converges for some fixed z=z0z=z_{0}, then it converges absolutely for every zz satisfying z<z0|z|<\left|z_{0}\right|.

Define the radius of convergence of a power series.

Give an example of vCv \in \mathbb{C} and an example of wCw \in \mathbb{C} such that v=w=1,n=1vnn|v|=|w|=1, \sum_{n=1}^{\infty} \frac{v^{n}}{n} converges and n=1wnn\sum_{n=1}^{\infty} \frac{w^{n}}{n} diverges. [You may assume results about standard series without proof.] Use this to find the radius of convergence of the power series n=1znn\sum_{n=1}^{\infty} \frac{z^{n}}{n}.

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