Chapter 8: Q- 17E (page 434)

Question: In Problems 17-20, find a power series expansion for f'(C), given the expansion for f(x).
17. $f (x) = {(1 + x)}^{- 1} = \sum_{n = 0}^{m} (- 1^{n} x^{n}$

Short Answer

Expert verified

The differentiation for the series $f (x) = \sum_{n = 0}^{\infty} {a_{n}^{X}}^{n}$ is given as,

$f' (x) = \sum_{n = 0}^{\infty} n ({a_{n}^{X}}^{n - 1})$

Step by step solution

Differentiation of given series

The differentiation for the series $f (x) = \sum_{n = 0}^{\infty} {a_{n}^{X}}^{n}$ is given as,

$f' (x) = \sum_{n = 0}^{\infty} n ({a_{n}^{X}}^{n - 1})$

Differentiate the given power series

The given series is

$f' x = {(1 + x)}^{- 1} = \sum_{n = 0}^{m} {(- 1)}^{n} x^{n}$

Find the differentiation of the above as,

$f' x = \sum_{n = 0}^{m} (n {(- 1)}^{n} {x^{n}}^{- 1})$

Hence, the required power series expansion is $f' x = \sum_{n = 0}^{m} (n {(- 1)}^{n} {x^{n}}^{- 1})$ .

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Question: In Problems 17-20, find a power series expansion for f'(C), given the expansion for f(x).
17. $f (x) = {(1 + x)}^{- 1} = \sum_{n = 0}^{m} (- 1^{n} x^{n}$

Short Answer

Step by step solution

Differentiation of given series

Differentiate the given power series

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Question: In Problems 17-20, find a power series expansion for f'(C), given the expansion for f(x).17.fx=1+x-1=∑n=0m(-1nxn

Short Answer

Step by step solution

Differentiation of given series

Differentiate the given power series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Question: In Problems 17-20, find a power series expansion for f'(C), given the expansion for f(x).
17. $f (x) = {(1 + x)}^{- 1} = \sum_{n = 0}^{m} (- 1^{n} x^{n}$