Q- 17E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Question: In Problems 17-20, find a power series expansion for f'(C), given the expansion for f(x) .

17. $f\left(x\right)={\left(1+x\right)}^{-1}=\sum _{n=0}^m(-1^n{x}^n$

Step-by-Step Solution

Answer

  

Step 1: Differentiation of given series

The differentiation for the series$f\left(x\right)=\sum _{n=0}^{\infty }{a_n^X}^n$is given as,

$f\text{'}\left(x\right)=\sum _{n=0}^{\infty }n\left({a_n^X}^{n-1}\right)$

Step 2: Differentiate the given power series

The given series is

$f\text{'}x={\left(1+x\right)}^{-1}=\sum _{n=0}^m{(-1)}^n{x}^n$

Find the differentiation of the above as,

 $f\text{'}x=\sum _{n=0}^m\left(n{(-1)}^n{x^n}^{-1}\right)$

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