Paper 1, Section II, F

Analysis I | Part IA, 2021

(a) Let n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n} be a power series with anCa_{n} \in \mathbb{C}. Show that there exists R[0,]R \in[0, \infty] (called the radius of convergence) such that the series is absolutely convergent when z<R|z|<R but is divergent when z>R|z|>R.

Suppose that the radius of convergence of the series n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n} is R=2R=2. For a fixed positive integer kk, find the radii of convergence of the following series. [You may assume that limnan1/n\lim _{n \rightarrow \infty}\left|a_{n}\right|^{1 / n} exists.] (i) n=0ankzn\sum_{n=0}^{\infty} a_{n}^{k} z^{n}. (ii) n=0anzkn\sum_{n=0}^{\infty} a_{n} z^{k n}. (iii) n=0anzn2\sum_{n=0}^{\infty} a_{n} z^{n^{2}}.

(b) Suppose that there exist values of zz for which n=0bnenz\sum_{n=0}^{\infty} b_{n} e^{n z} converges and values for which it diverges. Show that there exists a real number SS such that n=0bnenz\sum_{n=0}^{\infty} b_{n} e^{n z} diverges whenever Re(z)>S\operatorname{Re}(z)>S and converges whenever Re(z)<S\operatorname{Re}(z)<S.

Determine the set of values of zz for which

n=02neinz(n+1)2\sum_{n=0}^{\infty} \frac{2^{n} e^{i n z}}{(n+1)^{2}}


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