   Chapter 11.8, Problem 41E

Chapter
Section
Textbook Problem

# Suppose the series ∑ cnxn has radius of convergence 2 and the series ∑ dnxn has radius of convergence 3. What is the radius of convergence of the series ∑(cn + dn)xn?

To determine

To find: The radius of convergence of the series, (cn+dn)xn if the series cnxn has radius of convergence 2 and the series dnxn has radius of convergence 3.

Explanation

Let x0(2,2) be fixed.

Since the radius of convergence of cnxn is 2 and the radius of convergence of dnxn is 3, cn(x0)n and dn(x0)n are convergent.

Let cn(x0)n=C and dn(x0)n=D , then their sum (cn+dn)(x0)n is computed as follows,

(cn+dn)(x0)n=cn(x0)n+dn(x0)n=C+D

Since x0(2,2) , the radius of convergence of (cn+dn)xn is less or equal to 2.

Suppose that the radius of convergence of the series (cn+dn)xn is r>2 .

Then, the series (cn+dn)xn is convergent for every x(r,r)

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