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 $\lim_{n \to \infty} R_{n} (x) = 0$ on the specified interval.
$e^{x}, 1, ℝ$

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 $P_{n} (x) = \sum_{k = 0}^{\infty} \frac{e}{k!} (x - 1)^{k}$

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

Calculation

For $n \geq 0, f^{(x + 1)} (x)$ 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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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→∞Rn(x)=0on the specified interval.ex,1,ℝ

Short Answer

Step by step solution

Given Information

Calculation

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Most popular questions from this chapter

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Calculus

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Geometry

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