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Chapter 9: Problem 5

How is the remainder \(R_{n}(x)\) in a Taylor polynomial defined?

Short Answer

Expert verified
Answer: The remainder term, denoted as \(R_{n}(x)\), represents the error associated with approximating a function using a Taylor polynomial of degree 'n'. According to Taylor's theorem, for a smooth function \(f(x)\) and its Taylor polynomial approximation \(T_n(x)\), there exists a remainder term \(R_n(x)\) such that \(f(x) = T_n(x) + R_{n}(x)\). The theorem also states that there is a point \(\xi\) in the interval between the expansion point and 'x', for which \(R_{n}(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{(n+1)}\). The remainder term helps quantify the accuracy of the Taylor polynomial approximation and depends on the degree of the polynomial. An increase in the degree generally results in a smaller error and better approximation.

Step by step solution

01

Understand Taylor Polynomials

Taylor polynomial is a polynomial that approximates a function. It's an infinite sum of terms derived from the function's derivatives at a single point. The Taylor polynomial's degree determines how accurate the approximation will be. The more terms included, the more accurate the approximation.
02

Examine the Degree of Taylor Polynomials

The degree of a Taylor polynomial is represented as 'n'. It represents the maximum power of x in the polynomial, and thus, the number of terms considered in the polynomial. Higher values of 'n' generally result in a more accurate approximation of the function.
03

Introduction to Remainder Term \(R_{n}(x)\)

When using a Taylor polynomial to approximate a function, there is always an error associated with the approximation. This error is called the remainder term, denoted as \(R_{n}(x)\). The remainder term quantifies the difference between the actual function and its Taylor polynomial approximation.
04

Taylor's Theorem and the Remainder Term

Taylor's theorem states that for a smooth function \(f(x)\) and its Taylor polynomial approximation \(T_n(x)\) of degree 'n', there exists a remainder term \(R_n(x)\) such that: $$f(x) = T_n(x) + R_{n}(x)$$ The theorem guarantees that, for a specific 'n', there exists a point \(\xi\) in the interval between the expansion point and 'x', such that: $$R_{n}(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{(n+1)}$$
05

Interpret the Remainder Term

In the expression for the remainder term, \(f^{(n+1)}(\xi)\) represents the (n+1)-th derivative of the function evaluated at the point \(\xi\), and \((n+1)!\) is the factorial of (n+1). The term \((x-a)^{(n+1)}\) indicates how the error depends on the distance between the expansion point 'a' and the point 'x' where the function is being approximated. The relationship between the remainder term and the Taylor polynomial shows that increasing the degree of the polynomial will generally result in a smaller error and a better approximation of the function.

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Most popular questions from this chapter

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. $$f(x)=\frac{e^{x}+e^{-x}}{2}$$

Inverse hyperbolic sine The inverse hyperbolic sine is defined in several ways; among them are $$ \sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}} $$ Find the first four terms of the Taylor series for \(\sinh ^{-1} x\) using these two definitions (and be sure they agree).

Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k}$$

Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) $$\sum_{k=0}^{\infty} e^{-k x}$$

Exponential function In Section 9.3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty< x <\infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=e^{-x}$$
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