Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If f has a Taylor series that converges only on (−2,2), then f(x²) has a Taylor series that also converges only on (−2,2).

Verified step by step guidance

1. Recall that the radius of convergence of a Taylor series centered at 0 is the distance from 0 to the nearest singularity of the function in the complex plane.
2. Given that the Taylor series of \( f \) converges only on the interval \( (-2, 2) \), this means the radius of convergence \( R \) of \( f \) is 2.
3. Now consider the function \( g(x) = f(x^2) \). To find the radius of convergence of the Taylor series of \( g \), analyze how the substitution \( x^2 \) affects the domain.
4. Since \( g(x) = f(x^2) \), the series for \( g \) converges when \( |x^2| < 2 \), which simplifies to \( |x| < \sqrt{2} \).
5. Therefore, the radius of convergence of the Taylor series for \( g(x) \) is \( \sqrt{2} \), which is different from the original radius 2, so the statement is false.

Key Concepts

Radius of Convergence of a Taylor Series

The radius of convergence is the distance from the center point within which a Taylor series converges to the function. It depends on the function's behavior and singularities in the complex plane. Understanding this helps determine where the series representation is valid.

Composition of Functions and Its Effect on Convergence

When composing functions, such as considering f(x²), the domain and convergence properties can change. The substitution x² affects the input range and may alter the radius of convergence of the resulting Taylor series.

Counterexamples in Analysis

Counterexamples demonstrate that a general statement is false by providing a specific case where it fails. Constructing or identifying counterexamples is crucial to test claims about convergence and function behavior rigorously.