EXERCISE 2.1. Suppose (an) and (bn) are sequences of complex numbers, and the series Σo anz" and Σobn" have radii of convergence R₁ and R₂, respectively. Show that the radius of convergence R of the Cauchy product of these two series satisfies R≥ min{R₁, R₂}. Give an example of two series where strict inequality holds, R > min{R₁, R₂}. (Hints: Try using Mertens' Theorem for the first part; don't overthink the last part.)
EXERCISE 2.1. Suppose (an) and (bn) are sequences of complex numbers, and the series Σo anz" and Σobn" have radii of convergence R₁ and R₂, respectively. Show that the radius of convergence R of the Cauchy product of these two series satisfies R≥ min{R₁, R₂}. Give an example of two series where strict inequality holds, R > min{R₁, R₂}. (Hints: Try using Mertens' Theorem for the first part; don't overthink the last part.)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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![EXERCISE 2.1. Suppose (an) and (bn) are sequences of complex numbers, and the series
Σno anz" and no bnz" have radii of convergence R₁ and R₂, respectively. Show that the radius of
convergence R of the Cauchy product of these two series satisfies R≥ min{R₁, R₂}. Give an example
of two series where strict inequality holds, R > min{R₁, R₂}. (Hints: Try using Mertens' Theorem for
the first part; don't overthink the last part.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fffbf47e9-5bb9-4b19-a28b-799790ab70ec%2Ffff1449b-95c5-4720-971e-85952279116f%2F2mpczj9_processed.png&w=3840&q=75)
Transcribed Image Text:EXERCISE 2.1. Suppose (an) and (bn) are sequences of complex numbers, and the series
Σno anz" and no bnz" have radii of convergence R₁ and R₂, respectively. Show that the radius of
convergence R of the Cauchy product of these two series satisfies R≥ min{R₁, R₂}. Give an example
of two series where strict inequality holds, R > min{R₁, R₂}. (Hints: Try using Mertens' Theorem for
the first part; don't overthink the last part.)
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