Chapter 8.7, Problem 1E

To determine

To find: The formula for $b_8$ .

Answer to Problem 1E

The required formula is $b_8=\frac{f^8\left(5\right)}{8!}$ .

Explanation of Solution

Given:

If $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{b}_n{\left(x-5\right)}^n}$ for all the value of $x$ and write the formula for $b_8$ .

Calculation:

Consider the given function is,

  $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{b}_n{\left(x-5\right)}^n}$

Consider that if $f$ has the power series representation at $a$ that is $f\left(x\right){\displaystyle \sum _{n=0}^{\infty }{b}_n{\left(x-a\right)}^n}$ , then the coefficients are given by the formula $b_n=\frac{f^{n}a}{n!}$ .

Then,

  $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{f^{n}a}{n!}{\left(x-a\right)}^n}$

Then,

  $\begin{array}{c}f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{b}_n{\left(x-5\right)}^n}\\ ={\displaystyle \sum _{n=0}^{\infty }\frac{f^n\left(5\right)}{n!}{\left(x-5\right)}^n}\end{array}$

Then, from the general function and the above function,

  $\begin{array}{l}a=5\\ b_n=\frac{f^{″}\left(n\right)}{n!},\frac{f^{″}\left(5\right)}{n!}\\ b_8=\frac{f^8\left(5\right)}{8!}\end{array}$