Paper 1, Section I, 4D4 \mathrm{D}

Analysis I | Part IA, 2018

Define the radius of convergence RR of a complex power series anzn\sum a_{n} z^{n}. Prove that anzn\sum a_{n} z^{n} converges whenever z<R|z|<R and diverges whenever z>R|z|>R.

If anbn\left|a_{n}\right| \leqslant\left|b_{n}\right| for all nn does it follow that the radius of convergence of anzn\sum a_{n} z^{n} is at least that of bnzn\sum b_{n} z^{n} ? Justify your answer.

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