Complex Analysis: Theory and Applications · Problem 4.4.8

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · radius-of-convergence-derivative-series

Problem

This exercise establishes that the radius of convergence for gg given in Theorem 4.17 is ρ\rho, the same as that of the function ff. ((a)) Explain why the radius of convergence for gg is 1lim⁡n→∞sup⁡∣ncn∣1n−1\dfrac{1}{\lim_{n\to\infty}\sup |nc_n|^{\frac{1}{n-1}}}. ((b)) Show that lim⁡n→∞sup⁡∣n∣1n−1=1\lim_{n\to\infty}\sup |n|^{\frac{1}{n-1}} = 1. Hint: The lim sup⁡\limsup equals the limit. Show that lim⁡n→∞log⁡nn−1=0\lim_{n\to\infty} \frac{\log n}{n-1} = 0. ((c)) Assuming that lim⁡n→∞sup⁡∣cn∣1n−1=lim⁡n→∞sup⁡∣cn∣1n\lim_{n\to\infty}\sup |c_n|^{\frac{1}{n-1}} = \lim_{n\to\infty}\sup |c_n|^{\frac{1}{n}}, show that the conclusion for this exercise follows. ((d)) Verify the truth of the assumption made in part (c).

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