Analysis | Part IA, 2006

Let n=0anzn\sum_{n=0}^{\infty} a_{n} z^{n} and n=0bnzn\sum_{n=0}^{\infty} b_{n} z^{n} be power series in the complex plane with radii of convergence RR and SS respectively. Show that if RSR \neq S then n=0(an+bn)zn\sum_{n=0}^{\infty}\left(a_{n}+b_{n}\right) z^{n} has radius of convergence min(R,S)\min (R, S). [Any results on absolute convergence that you use should be clearly stated.]

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