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 11.3.7

Chapter 11, Problem 11.3.7

In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f?

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Recall that the Taylor series of a function \(f\) centered at a point \(a\) is given by the infinite sum \(\displaystyle \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n\).

The remainder term \(R_n(x)\) represents the error between the function \(f(x)\) and the \(n\)th partial sum of its Taylor series, i.e., \(R_n(x) = f(x) - \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k\).

For the Taylor series to converge to \(f(x)\) at a point \(x\), the remainder \(R_n(x)\) must approach zero as \(n\) approaches infinity: \(\lim_{n \to \infty} R_n(x) = 0\).

This means that the partial sums of the Taylor series get arbitrarily close to the actual function value \(f(x)\) as more terms are included.

Therefore, convergence of the Taylor series to \(f\) in terms of the remainder means the remainder term becomes negligible, ensuring the infinite series accurately represents the function at that point.

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Key Concepts

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

Taylor Series and Its Remainder

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. The remainder is the difference between the actual function value and the partial sum of the series up to a certain degree, measuring the error in approximation.

Convergence of a Taylor Series

Convergence means that as more terms are added, the Taylor series sum approaches the actual function value. For the series to converge to the function, the remainder must approach zero as the number of terms increases indefinitely.

Role of the Remainder in Function Approximation

The remainder quantifies how closely the Taylor polynomial approximates the function. If the remainder tends to zero for all points in the interval, the Taylor series converges to the function, ensuring the series accurately represents the function within that domain.

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Related Practice

Textbook Question

Differential equations

a. Find a power series for the solution of the following differential equations, subject to the given initial condition

b. Identify the function represented by the power series.

y′(t) − y = 0, y(0) = 2

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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) = 1/(1 - x), a=0

Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x sin(1/x)

Textbook Question

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.

g(x) = x/(1 + x²)² using f(x) = 1/(1 + x²)

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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∞ (k²⁰ xᵏ)/(2k+1)!

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Textbook Question

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

1/(1 − 2x)

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