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.
32. f(x)=ln(1+x), x₀ =0

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

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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,

$f (x - x_{0}) = f (x_{0}) + f' (x_{0}) . (x - x_{0}) + f'' (x_{0}) . \frac{{(x - x_{0})}^{2}}{2!} + f''' (x_{0}) . \frac{{(x - x_{0})}^{3}}{3!} + . . .$

Step 2: Derivatives of function at x0

We have to calculate the Taylor series expansion for,f (x)= ln (1 + x) at.

Calculating the derivatives of function at x₀,

f (x) =ln (1+x) then f (x₀)=0

f'(x) = $\frac{1}{1 + x}$ then f '(x₀) =1

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

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

f''''(x) = $\frac{- 6}{{(1 + x)}^{4}}$ then f''''(x₀) = -6

Step 3: Substitute the derivatives in Taylor series

Substituting the above derivatives in Taylor series expansion for the function at^{$x_{0} = 0$ ,}then,

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

= $x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + . . .$

= role="math" localid="1664200514125" $\sum_{n - 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n} . {(x)}^{n}$

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

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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.
32. f(x)=ln(1+x), x₀ =0

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 X0.32. f(x)=ln(1+x), x0 =0

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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Question: In Problems 29–34, determine the Taylor series about the point X₀ for the given functions and values of X₀.
32. f(x)=ln(1+x), x₀ =0