Chapter 8: Q. 49 (page 692)

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.
localid="1649857668603" $\ln (x), 3, (2, 4)$

Short Answer

Expert verified

The Lagrange's form for the remainder $R_{n} (x)$ on the specified interval is $|R_{n} (x)| = \frac{1}{(n + 1) c^{n + 1}} {(x - 3)}^{n + 1}$

Step by step solution

Step 1. Given Information

The function is, $f (x) = \ln (x)$

Phow that $\lim_{n \to \infty} R_{n} (x) = 0$ on the specified interval.

Step 2. Calculation

If $f (x) = \ln (x)$ , we know that for every $n \geq 0, |f^{n + 1} (c)| \leq 1$ , for every value of $x$ , so using the Lagrange's form for the remainder, we have

$|R_{n} (x)| = \frac{f^{(n + 1)} (c)}{(n + 1)!} {(x - x_{0})}^{n + 1}$

Step 3. Simplification

Since the Taylor series for the function $f (x) = \ln (x)$ , at $x = 3$ is

$P_{n} (x) = 1 + \frac{1}{2} (x - 1) + \sum_{k = 2}^{\infty} {(- 1)}^{k + 1} \frac{1.2.3 . . . (k - 1)}{3}$ ^kk!(x-3)k

Therefore, $|R_{n} (x)| = \frac{1}{(n + 1) c^{n + 1}} {(x - 3)}^{n + 1}$

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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.
localid="1649857668603" $\ln (x), 3, (2, 4)$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Simplification

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You were asked to find the Taylor series for the specified function at the given value of x0. In Exercises 45-50 find the Lagrange's form for the remainder Rn(x), and show that limn→∞Rn(x)=0on the specified interval.localid="1649857668603" lnx,3,(2,4)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Geometry

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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.
localid="1649857668603" $\ln (x), 3, (2, 4)$