Chapter 9.2, Problem 43E

$\text{(a)}$

To determine

To find: The power series.

Answer to Problem 43E

The answer is:

Explanation of Solution

Given information:

The function is $f(x)=(\sin x)/x$ .

Calculation:

The power series can be obtained from the Maclaurin series of $\sin x$ by dividing $x$:

  

Therefore, the required power series is

$\text{(b)}$

To determine

To explain: The power series part $\text{(a)}$ is not quite a Maclaurin series for $f$ .

Answer to Problem 43E

The function in part $\text{(a)}$ is undefined at $x=0$ , hence so is the obtained Maclaurin series.

Explanation of Solution

Given information:

The function is $f(x)=(\sin x)/x$ .

Calculation:

The method employed in part $\text{(a)}$ has a flaw in that it only works if the result after division by $x$ is not zero. The Quotient Rule can be used to demonstrate that the $f$ function and its derivatives are undefined at $x=0$ .

The Maclaurin series for $f$ in part $\text{(a)}$ is ineligible since, by definition, $x$ should be zero in this situation.

$(c)$

To determine

To find: The value of $k$ .

Answer to Problem 43E

The value of $k$ is $1$ .

Explanation of Solution

Given information:

The function is $f(x)=(\sin x)/x$ .

Calculation:

The value of $k$ must be such that the function is well approximated by the Maclaurin series. The function must also be continuous since it must be differentiable on the interval containing zero.

The function has a detachable discontinuity even though it is undefined $x=0$:

  

In other words, as $x$ approaches $0$ , then $f(x)$ approaches $1$ , hence $k$ must be $1$ .