Q.71 - Calculus

Question

Use appropriate Maclaurin series to express the quantities in Exercises $67-76$ as alternating series. Then use Theorem$7.38$ to approximate the value of the specified quantities to within $0.001$ of their actual value. How many terms in each series would be needed to approximate the given quantity to within ${10}^{-6}$ of its value? In Exercises $73-76$ be sure to convert to radian measure first.

${\tan }^{-1}0.4$

Step-by-Step Solution

Answer

the approximate the value of  ${\tan }^{-1}0.4$ up to three decimal places is  $0.409$

Also, to approximate the quantity $ta{n}^{-1}0.4$ to within ${10}^{-6}$ of its value, the number of terms required is $14$

step 1: given information

Consider the function ${\tan }^{-1}0.4$

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

The Maclaurin series for the function $f\left(x\right)={\tan }^{-1}x$ is

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

step 2: caluculation

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

The Maclaurin series for the function $f\left(x\right)={\tan }^{-1}x$ is

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

So, to find the Maclaurin series for the function $ta{n}^{-1}0.4$, put $x=0.4$Therefore,

$f(x=0.4)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{2k+1}(0.4{)}^{2k+1}$

that is,

${\tan }^{-1}0.4=\sum _{k=1}^{\infty }\frac{(-1{)}^{k+1}}{k}(0.4{)}^k$

Now, to approximate the value of ${\tan }^{-1}0.4$ up to three decimal places, let us first write its corresponding Maclaurin series in expanded form.

So,

$$\begin{gathered} {\tan }^{-1}0.4=\sum _{k=1}^{\infty }\frac{(-1{)}^{k+1}}{k}(0.4{)}^d \\ =0.4-\frac{1}{2}(0.4{)}^2-\frac{1}{3}(0.4{)}^3+\frac{1}{4}(0.4{)}^4-\frac{1}{5}(0.4{)}^5 \\ =0.4-\frac{0.16}{2}+\frac{0.064}{3}-\frac{0.0256}{4}+\frac{0.01024}{5} \\ =0.4-0.008+0.0213-0.0064+0.002048 \end{gathered}$$

Therefore, the approximate the value of ${\tan }^{-1}0.4$ up to three decimal places is $0.409$

Also, to approximate the quantity ${\tan }^{-1}0.4$ to within ${10}^{-6}$ of its value, the number of terms required is 14