Chapter 9, Problem 63RE

$\text{(a)}$

To determine

To find: The power series.

Answer to Problem 63RE

The answer:

Explanation of Solution

Given information:

The integral is

Calculation:

  

Therefore, the required power series is

$\text{(b)}$

To determine

To find: The number of terms of the series.

Answer to Problem 63RE

The answer: At least two terms of the series.

Explanation of Solution

Given information:

The integral is

Calculation:

In part $(a)$ at $x=1$ ,

in this case,

  

The error of the approximation for $n$ is:

  $\text{error }<\frac{1}{(4n+3)(2n+1)!}$ which must be less than $0.001$ .

If,

  $\begin{array}{c}\frac{1}{(4n+3)(2n+1)!}<0.001\\ 1000<(4n+3)(2n+1)!\end{array}$

From which

  $n\ge 2$

because

  $\begin{array}{c}(4\cdot 2+3)(2\cdot 2+1)!=11\cdot 5!\\ =1320\end{array}$

Based on this, should use at least two terms of the series for the approximation with an error less than $0.001$ .

Therefore, the required at least two terms of the series.

$(c)$

To determine

To find: The value of given integral.

Answer to Problem 63RE

The answer:

Explanation of Solution

Given information:

The integral is

Calculation:

From part $(b)$:

  

The first five partial sums of the series are:

  

Therefore, the required value of

$(d)$

To determine

To find: The difference between of the exact value and the approximation if use four terms of the series in part $(b)$ .

Answer to Problem 63RE

The answer: The difference between of the exact value and the approximation is approximately $1.45\cdot {10}^{-7}$ if use four terms of the series in part $(b)$ .

Explanation of Solution

Given information:

The integral is

Calculation:

The error of the approximation is less than the absolute value of the series' $5$ th term, according to the Alternating Series Bound Theorem.

  $\begin{array}{c}\text{ error }<\frac{1}{(4\cdot 4+3)(2\cdot 4+1)!}=\frac{1}{19\cdot 9!}\\ \approx 1.45\cdot {10}^{-7}\end{array}$

Therefore, the required difference between of the exact value and the approximation is approximately $1.45\cdot {10}^{-7}$ if use four terms of the series in part $(b)$ .