Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function f(x) = √x has a Taylor series centered at 0.

Verified step by step guidance

1. Recall that a Taylor series of a function \(f(x)\) centered at \(a\) is given by the infinite sum \(\displaystyle \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n\), where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(a\).
2. For the function \(f(x) = \sqrt{x} = x^{1/2}\), consider the point \(a = 0\) where the Taylor series is centered. We need to check if all derivatives \(f^{(n)}(0)\) exist and are finite.
3. Calculate the first derivative: \(f'(x) = \frac{1}{2} x^{-1/2}\). Notice that as \(x \to 0^+\), \(f'(x)\) tends to infinity, so \(f'(0)\) is not defined.
4. Since the first derivative at \(x=0\) does not exist, the Taylor series centered at 0 cannot be formed because the coefficients \(\frac{f^{(n)}(0)}{n!}\) are not all defined.
5. Therefore, the function \(f(x) = \sqrt{x}\) does not have a Taylor series centered at 0. This is because the function is not differentiable at 0 in the usual sense required for a Taylor series expansion.

Key Concepts

Taylor Series and Its Definition

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. It requires the function to be infinitely differentiable at that point, and the series converges to the function within some interval around the center.

Differentiability of f(x) = √x at x = 0

The function f(x) = √x is defined for x ≥ 0, but its derivative f'(x) = 1/(2√x) becomes unbounded as x approaches 0 from the right. This means f(x) is not differentiable at 0 in the usual sense, which affects the existence of a Taylor series centered at 0.

Radius of Convergence and Analyticity

For a Taylor series to represent a function, the function must be analytic at the center point, meaning it can be expressed as a convergent power series there. Since f(x) = √x is not analytic at 0 due to the non-differentiability, its Taylor series centered at 0 does not exist or does not converge to the function.