Chapter 8.6, Problem 22E

To determine

To Find: The power series representation for $f$ and the graph f and several partial sums $s_n\left(x\right)$ on the screen and what happens as n increases.

Answer to Problem 22E

The required sketch is shown in Figure 1, the power series for $f\left(x\right)$ is $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n{\left(2x\right)}^{2n+1}}{\left(2n+1\right)}}$ and other power series is obtained.

Explanation of Solution

Given:

The given equation is $f\left(x\right)={\tan }^{-1}\left(2x\right)$ .

Calculation:

Consider the given function as,

  $f\left(x\right)={\tan }^{-1}\left(2x\right)$

Then,

  $\begin{array}{c}{\tan }^{-1}\left(2x\right)={\displaystyle \int \frac{2}{1+4x^2}dx}\\ ={\displaystyle \int {\displaystyle \sum _{n=0}^{\infty }2{\left(-4x^2\right)}^n}dx}\\ ={\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{2^{2n+1}{x}^{2n+1}}{\left(2n+1\right)}}+C\end{array}$

For the value of $x=0$ , $f\left(x\right)=0$ .

Then, $0=C$ .

Then,

  $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n{\left(2x\right)}^{2n+1}}{\left(2n+1\right)}}$

The series converges as $\left|-4x^2\right|<1\left|x\right|<\frac{1}{2}$ . Then, the interval is,

  $\left|-4x^2\right|<1\left|x\right|<\frac{1}{2}$

The term of the series is,

  $\begin{array}{l}{a}_0=2x\\ a_1=\frac{-{\left(2x\right)}^3}{3}\\ a_2=\frac{{\left(2x\right)}^5}{5}\\ a_3=-\frac{{\left(2x\right)}^7}{7}\end{array}$

Then,

  $a_4=\frac{{\left(2x\right)}^9}{7}$

Then, the sketch for the graph for $f\left(x\right)$ and $s_n\left(x\right)$ on the same screen as n increases $s_n\left(x\right)$ then, the better approximation for $f\left(x\right)$ as shown in Figure 1

  

Figure 1