Skip to main content
Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.3.67a
Chapter 11, Problem 11.3.67a

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 \(a\).
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.
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.
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.
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
08:42
Taylor Series

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.
Recommended video:
04:56
Derivative of the Natural Exponential Function (e^x)

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.
Recommended video:
07:36
Radius of Convergence
Related Practice
Textbook Question

Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.


a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.

Textbook Question

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.


a. √(1 + 2x)


A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

1
views
Textbook Question

Taylor series


a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.


f(x)=sin x, a = π/2

1
views
Textbook Question

Taylor series and interval of convergence


a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.


f(x)=2/(1−x)³, a=0

Textbook Question

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 


a. Estimate f(0.1) and give a bound on the error in the approximation.


f(x) = eˣ ≈ 1 + x

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. Only even powers of x appear in the Taylor polynomials for f(x)=e⁻²ˣ centered at 0.

1
views