Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Question: In Problems 29–34, determine the Taylor series about the point x₀ for the given functions and values of x₀ .
34. f(x)= $\sqrt{x,} x_{0} = 1$

The required expression is $1 + \frac{1}{2} (X - 1) + \sum_{n - 2}^{\infty} \frac{{(- 1)}^{n + 1} (2 n - 3)!}{2^{2 n - 2} (n - 2)! (n)!} {(x - 1)}^{n}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Taylor series

For a function ^f(x) the Taylor series expansion about a point x₀ is given by, $f (x - x_{0}) = f (x_{0}) + f' (x_{0}) . (x - x_{0}) + f " (x_{0}) - \frac{(x - x_{0})}{2!} + f''' (x_{0}) - \frac{{(x - x_{0})}^{3}}{3!} + . . . .$

Step 2: Derivatives of function at x0

We have to calculate the Taylor series expansion for, $f (x) = \sqrt{x}$ at x₀=1 .

Calculating the derivatives of function at x₀ .

f (x) = $\sqrt{x}$ then f (x₀) =1

f'(x) = $\frac{1}{2} x^{- 1 / 2}$ then f'(x₀) = $\frac{1}{2}$

f''(x) = $\frac{- 1}{4} x^{- 3 / 2}$ then f''(x₀) = $\frac{- 1}{4}$

f'''(x) = $\frac{3}{8} x^{- 5 / 2}$ then f'''(x₀) = $\frac{3}{8}$

f''''(x) = $\frac{- 15}{16} x^{- 7 / 2}$ then f''''(x₀) = $\frac{- 15}{16}$

Step 3: Substitute the derivatives in Taylor series

Substituting the above derivatives in Taylor series expansion for the function at x₀=1, then,

$\sqrt{x} = 1 + \frac{1}{2} (x - 1) - \frac{1}{4} \frac{{(x - 1)}^{2}}{2!} + \frac{3}{8} \frac{{(x - 1)}^{3}}{3!} - \frac{15}{16} \frac{{(x - 1)}^{4}}{4!} + . . . .$

= $1 + \frac{1}{2} (x - 1) + \sum_{n - 2}^{\infty} \frac{{(- 1)}^{n + 1}}{2^{n}} \frac{(2 n - 3)!}{2^{n} 2^{n - 2} (n - 2)! (n)!} {(x - 1)}^{n}$

= role="math" localid="1664103079471" $1 + \frac{1}{2} (x - 1) + \sum_{n - 2}^{\infty} \frac{{(- 1)}^{n + 1} (2 n - 3)!}{2^{2 n - 2} (n - 2)! (n)!} {(x - 1)}^{n}$

Hence, the required expression is $1 + \frac{1}{2} (x - 1) + \sum_{n - 2}^{\infty} \frac{{(- 1)}^{n + 1} (2 n - 3)!}{2^{2 n - 2} (n - 2)! (n)!} {(x - 1)}^{n}$

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Question: In Problems 29–34, determine the Taylor series about the point x₀ for the given functions and values of x₀ .
34. f(x)= $\sqrt{x,} x_{0} = 1$

Step by Step Solution

Step 1: Taylor series

Step 2: Derivatives of function at x0

Step 3: Substitute the derivatives in Taylor series

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Question: In Problems 29–34, determine the Taylor series about the point x0 for the given functions and values of x0 .34. f(x)=x, x0=1

Step by Step Solution

Step 1: Taylor series

Step 2: Derivatives of function at x0

Step 3: Substitute the derivatives in Taylor series

Most popular questions for Math Textbooks

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Question: In Problems 29–34, determine the Taylor series about the point x₀ for the given functions and values of x₀ .
34. f(x)= $\sqrt{x,} x_{0} = 1$