Chapter 8: Q. 48 (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.
$\sqrt{x}, 1, (1 / 2, 3 / 2)$

Short Answer

Expert verified

The required Lagrange's form for the given remainder is, $|R_{n} (x)| = {\frac{|x - π|}{(n + 1)!}}^{n + 1}$

Step by step solution

Step 1. Given Information

The function is $f (x) = \sqrt{x}$

. Calculation

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

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

Step 3. Simplification

Since the Taylor series for the function $f (x) = \sqrt{x}$ at $x = 1$ is

localid="1649858569361" $P_{n} (x) = 1 + \frac{1}{2} (x - 1) + \sum_{k = 2}^{\infty} {(- 1)}^{k + 1} \frac{k . 1.3.5 . . . (2 k - 3)}{2}$ ^kk!(x-1)k

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

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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.
$\sqrt{x}, 1, (1 / 2, 3 / 2)$

Short Answer

Step by step solution

Step 1. Given Information

. Calculation

Step 3. Simplification

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Most popular questions from this chapter

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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.x,1,(1/2,3/2)

Short Answer

Step by step solution

Step 1. Given Information

. Calculation

Step 3. Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Statistics

Mechanics Maths

Logic and Functions

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.
$\sqrt{x}, 1, (1 / 2, 3 / 2)$