Book edition 9th

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

Pages 616 pages

ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Question: In Problems 29–34, determine the Taylor series about the point X₀ for the given functions and values of X₀.
30. $f (x) = x^{- 4} x_{0} = 1$

The required expression is . $\sum_{n - 0}^{\infty} {(- 1)}^{n} (x - 1)$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Taylor series

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

Step 2: Derivatives of function at x0.

We have to calculate the Taylor series expansion for, f (x) = x^-1 at x₀=1.

Calculating the derivatives of function at x₀.

$f (x) = x^{- 1} t h e n f (x_{0}) = 1$

$f' (x) = x^{- 2} t h e n f' (x_{0}) = - 1$

$f'' (x) = 2 x^{- 3} t h e n f'' (x_{0}) = 2$

$f''' (x) = 6 x^{- 4} t h e n f''' (x_{0}) = - 6$

$f'''' (x) = 24 x^{- 5} t h e n f'''' (x_{0}) = 24$

Step 3: Substitute the derivatives in Taylor series

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

${(x - 1)}^{- 1} = 1 - 1 (x - 1) + 2 . \frac{{(x - 1)}^{2}}{2!} - 6 \frac{{(x - 1)}^{3}}{3!} + 24 . \frac{{(x - 1)}^{4}}{4!} + . . . . = 1 - (x - 1) + {(x - 1)}^{2} - {(x - 1)}^{3} + {(x - 1)}^{4} + . . . . = \sum_{n - 0}^{\infty} {(- 1)}^{n} . {(x - 1)}^{n}$

Hence, the required expression is $= \sum_{n - 0}^{\infty} {(- 1)}^{n} . {(x - 1)}^{n}$

Find Study Materials for

Create Study Materials

Select your language

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.
30. $f (x) = x^{- 4} x_{0} = 1$

Step by Step Solution

Step 1: Taylor series

Step 2: Derivatives of function at x0.

Step 3: Substitute the derivatives in Taylor series

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

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 X0.30. f(x)=x-4 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

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Question: In Problems 29–34, determine the Taylor series about the point X₀ for the given functions and values of X₀.
30. $f (x) = x^{- 4} x_{0} = 1$