Chapter 8: Q-32E (page 435)

Question: In Problems 29–34, determine the Taylor series about the point X₀for the given functions and values of X₀.
32. f(x)=ln(1+x), x₀ =0

Short Answer

Expert verified

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

Step by step solution

Taylor series

For a function f(x) the Taylor series expansion about a point x₀ is given by,

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

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) = $\frac{1}{1 + x}$ thenf '(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

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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Question: In Problems 29–34, determine the Taylor series about the point X₀for the given functions and values of X₀.
32. f(x)=ln(1+x), x₀ =0

Short Answer

Step by step solution

Taylor series

Derivatives of function at x0

Substitute the derivatives in Taylor series

One App. One Place for Learning.

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Question: In Problems 29–34, determine the Taylor series about the point X0for the given functions and values of X0.32. f(x)=ln(1+x), x0 =0

Short Answer

Step by step solution

Taylor series

Derivatives of function at x0

Substitute the derivatives in Taylor series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Pure Maths

Calculus

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Question: In Problems 29–34, determine the Taylor series about the point X₀for the given functions and values of X₀.
32. f(x)=ln(1+x), x₀ =0