Q. 46 In Exercises 49-54聽in Section 8... [FREE SOLUTION]

Chapter 8: Q. 46 (page 692)

In Exercises $49 - 54$ in Section $8.2$ you were asked to find the Taylor series for the specified function at the given value of $x_{0}$ . In Exercises $45 - 50$ find the Lagrange's form for the remainder $R_{n} (x)$ , and show that $\lim_{n 鈫� 鈭�} R_{n} (x) = 0$ on the specified interval.
$e^{x}, 1, 鈩�$

Short Answer

Expert verified

The required Lagrange's form is $|R_{n} (x)| = \frac{e | x - 1 |^{n + 1}}{(n + 1)!}$

Step by step solution

Given Information

Consider the function $f (x) = e^{x}$ . The Taylor series of the function $f$ around $x = 1$ is $P_{n} (x) = {鈭�}_{k = 0}^{鈭�} \frac{e}{k!} (x - 1)^{k}$

The objective is to find the Lagrange's form for the remainder $R_{n} (x)$ .

Calculation

For $n 鈮� 0, f^{(x + 1)} (x)$ is $e^{x}$ .

The Lagrange's form for the remainder $R_{n} (x)$ is $R_{n} (x) = \frac{f^{(n + 1)} (c)}{(n + 1)!} x^{n + 1}$ . Thus,

$|R_{n} (x)| = |\frac{f^{(n + 1)} (c)}{(n + 1)!} x^{n + 1}|$

Therefore, the Lagrange's form is $|R_{n} (x)| = \frac{e | x - 1 |^{n + 1}}{(n + 1)!}$

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91影视

In Exercises 49-54in Section 8.2you were asked to find the Taylor series for the specified function at the given value of x0. In Exercises 45-50find the Lagrange's form for the remainder Rn(x), and show that limn鈫�鈭�Rn(x)=0on the specified interval.ex,1,鈩�

Short Answer

Step by step solution

Given Information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Pure Maths

Calculus

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.