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 11.3.65

Chapter 11, Problem 11.3.65

Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.

f(x) = e⁻ˣ, a = 0

Verified step by step guidance

Identify the function and the center of the Taylor series: here, the function is \(f(x) = e^{-x}\) and the series is centered at \(a = 0\).

Recall the Taylor series expansion formula for a function \(f(x)\) centered at \(a\): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n + R_n(x),\] where \(R_n(x)\) is the remainder after \(n\) terms.

Find the \(n\)-th derivative of \(f(x) = e^{-x}\). Since the derivative of \(e^{-x}\) is \(-e^{-x}\), the \(n\)-th derivative is \[f^{(n)}(x) = (-1)^n e^{-x}.\] Evaluate this at \(a=0\) to get \[f^{(n)}(0) = (-1)^n e^{0} = (-1)^n.\]

Write the remainder term \(R_n(x)\) using the Lagrange form of the remainder: \[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - 0)^{n+1} = \frac{(-1)^{n+1} e^{-c}}{(n+1)!} x^{n+1},\] where \(c\) is some number between \(0\) and \(x\).

To show that \(\lim_{n \to \infty} R_n(x) = 0\) for all \(x\) in the interval of convergence, analyze the absolute value: \[|R_n(x)| = \left| \frac{e^{-c}}{(n+1)!} x^{n+1} \right| \leq \frac{e^{-|c|}}{(n+1)!} |x|^{n+1}.\] Since \(e^{-c}\) is bounded and factorial growth in the denominator dominates any power of \(x\), the limit of \(R_n(x)\) as \(n \to \infty\) is zero for all real \(x\).

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Key Concepts

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

Taylor Series and Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point a. It approximates the function near a by polynomials, where each term involves higher-order derivatives evaluated at a, multiplied by powers of (x - a). Understanding this expansion is essential to express f(x) = e^{-x} around a = 0.

Remainder Term in Taylor Series

The remainder term Rₙ(x) measures the error between the actual function and its nth-degree Taylor polynomial approximation. It quantifies how well the polynomial approximates the function and is often expressed using Lagrange's form involving the (n+1)th derivative. Analyzing Rₙ(x) helps determine the accuracy and convergence of the series.

Limit of the Remainder and Interval of Convergence

Showing that limₙ→∞ Rₙ(x) = 0 within the interval of convergence proves that the Taylor series converges to the function f(x). The interval of convergence is the set of x-values where the series converges. For e^{-x}, the series converges for all real x, ensuring the remainder vanishes as n grows large.

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Related Practice

Textbook Question

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.

f(x) = sin x, a = 0

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Textbook Question

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

∑ₖ₌₀∞ (2x)ᵏ/k!

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Textbook Question

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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Textbook Question

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

−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯

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Textbook Question

{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.

e⁰ᐧ²⁵, n=4

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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) = cos x, a = π/4; approximate cos (0.24π)

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Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.f(x) = e⁻ˣ, a = 0

Verified video answer for a similar problem:

Key Concepts

Taylor Series and Expansion

Remainder Term in Taylor Series

Limit of the Remainder and Interval of Convergence

Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.

f(x) = e⁻ˣ, a = 0