Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.3.3

Chapter 11, Problem 11.3.3

Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.

Verified step by step guidance

Recall the general formula for the Taylor series of a function \(f\) centered at \(a\): \[f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x - a)^k,\] where \(f^{(k)}(a)\) is the \(k\)-th derivative of \(f\) evaluated at \(a\).

In this problem, the center is \(a = 2\), and we are given that for all nonnegative integers \(k\), \[f^{(k)}(2) = 1.\]

Substitute the given derivative values into the Taylor series formula: \[f(x) = \sum_{k=0}^{\infty} \frac{1}{k!} (x - 2)^k.\]

Recognize that this series matches the expansion of the exponential function \(e^y = \sum_{k=0}^{\infty} \frac{y^k}{k!}\), where \(y = x - 2\).

Therefore, the Taylor series for \(f\) centered at 2 can be expressed as \[f(x) = \sum_{k=0}^{\infty} \frac{(x - 2)^k}{k!}.\] This is the series representation of \(e^{x-2}\), but since the problem only asks for the Taylor series, this summation form is the key result.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. Centered at a point a, it approximates the function near a using powers of (x - a) weighted by the function's derivatives at a.

Derivatives and Their Role in Taylor Series

The coefficients of a Taylor series are determined by the function's derivatives at the center point. Specifically, the k-th term involves the k-th derivative evaluated at the center, divided by k factorial, multiplied by (x - a)^k.

Factorials and Series Coefficients

Factorials (k!) appear in the denominator of each Taylor series term to normalize the contribution of the k-th derivative. This ensures the series converges properly and accurately represents the function near the center.

Recommended video:

5:22

Factorials

Related Practice

Textbook Question

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

f(x) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)

views

Textbook Question

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = tan⁻¹(4x), a = 0

views

Textbook Question

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ² sin x² dx

views

Textbook Question

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

Find the Taylor polynomials p₁, …, p₅ centered at a=0 for f(x)=e⁻ˣ

views

Textbook Question

Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.

f(x) = eᶜᵒˢ ˣ

107

views

Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)

views