Question

Suppose that F(x) is a power series with radius of convergence R = 12. Which of the following is false?

A. F'(x)is a power series with radius of convergence R=12.
B. F(3x is a power series with radius of convergence R = 12/3 = 4.
C. F(x/3)isapower series with radius of convergence R=12/(1/3)= 36.
D. F(x)dx is apower series with radius of convergenceR=1.

Answer 1

The false statement is D. F(x)dx is a power series with radius of convergence R=1.

Step-by-step explanation:

In this question, we are given that the power series F(x) has a radius of convergence R = 12. We are asked to identify which of the following statements is false.

A. F'(x) is a power series with radius of convergence R = 12.

B. F(3x) is a power series with radius of convergence R = 12/3 = 4.

C. F(x/3) is a power series with radius of convergence R = 12/(1/3) = 36.

D. F(x)dx is a power series with radius of convergence R = 1.

Statement D is false. The radius of convergence of the derivative of a power series remains the same as the original series. Therefore, the correct answer is D.