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 11.3.1

Chapter 11, Problem 11.3.1

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

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Recall that the Taylor polynomial of degree n for a function \( f \) centered at \( a \) is given by the formula: \[ P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k \] where \( f^{(k)}(a) \) denotes the \( k \)-th derivative of \( f \) evaluated at \( a \).

Understand that the Taylor series of \( f \) centered at \( a \) is the infinite sum of all these terms, expressed as: \[ T(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} (x - a)^k \] This series represents the function \( f \) as an infinite polynomial expansion around \( a \).

Recognize that each Taylor polynomial \( P_n(x) \) is essentially the partial sum of the Taylor series up to degree \( n \). In other words, the Taylor polynomial approximates \( f \) by truncating the infinite series after \( n \) terms.

Note that as \( n \) increases, the Taylor polynomial \( P_n(x) \) generally provides a better approximation to \( f(x) \) near the point \( a \), and the Taylor series is the limit of these polynomials as \( n \to \infty \), assuming the series converges to \( f(x) \).

Therefore, the Taylor polynomials centered at \( a \) are the finite-degree approximations that build up to the full Taylor series centered at \( a \), which is the infinite-degree polynomial representation of \( f \) around that point.

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Key Concepts

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

Taylor Polynomial

A Taylor polynomial of degree n for a function f centered at a is a finite sum that approximates f near a using derivatives of f at a. It includes terms up to the nth derivative, providing a polynomial approximation that becomes more accurate as n increases.

Taylor Series

The Taylor series of a function f centered at a is an infinite sum of terms derived from the derivatives of f at a. It represents the function as a power series and, if convergent, equals the function within a certain interval around a.

Relationship Between Taylor Polynomials and Taylor Series

Taylor polynomials are partial sums of the Taylor series; each polynomial includes a finite number of terms from the series. As the degree of the polynomial increases, it approaches the Taylor series, which is the limit of these polynomials as the number of terms goes to infinity.

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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.

Find the Taylor polynomials p₁, p₂, and p₃ centered at a=1 for f(x)=x³.

Textbook Question

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₁∞ (x²ᵏ)/k

Textbook Question

{Use of Tech} Remainders Let

f(x) = ∑ₖ₌₀∞ xᵏ = 1/(1−x) and Sₙ(x) = ∑ₖ₌₀ⁿ⁻¹ xᵏ

The remainder in truncating the power series after n terms is Rₙ = f(x) − Sₙ(x), which depends on x.

a. Show that Rₙ(x) = xⁿ /(1−x).

b. Graph the remainder function on the interval |x| < 1, for n=1, 2, and 3 . Discuss and interpret the graph. Where on the interval is |Rₙ(x)| largest? Smallest?

c. For fixed n, minimize |Rₙ(x)| with respect to x. Does the result agree with the observations in part (b)?

d. Let N(x) be the number of terms required to reduce |Rₙ(x)| to less than 10⁻⁶. Graph the function N(x) on the interval |x|<1.

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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 (ln (1 + t))/t dt

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 polynomial p₃ centered at a=e for f(x)=ln x.

Textbook Question

{Use of Tech} Graphing Taylor polynomials

a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n=1 and n=2.

b. Graph the Taylor polynomials and the function.

f(x)=sin x, a=π/4