Chapter 11, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

1. a. Let p_n be the nth-order Taylor polynomial for f centered at 2. The approximation p₃(2.1) ≈ f(2.1) is likely to be more accurate than the approximation p₂(2.2) ≈ f(2.2).
2. b. If the Taylor series for f centered at 3 has a radius of convergence of 6, then the interval of convergence is [−3, 9].
3. c. The interval of convergence of the power series ${\displaystyle \sum c_k{x}^k}$could be (−7/3, 7/3).
4. d. The Maclaurin series for f(x) = (1 + x)¹² has a finite number of nonzero terms.
5. e. If the power series ${\displaystyle \sum c_k{(x-3)}^k}$has a radius of convergence of R = 4 and converges at the endpoints of its interval of convergence, then its interval of convergence is [−1, 7].

To determine

Whether the statement “Let $p_n$ be the nth-order Taylor polynomial for f centered at 2. The approximation $p_3\left(2.1\right)\approx f\left(2.1\right)$ is likely to be more accurate than the approximation $p_2\left(2.2\right)\approx f\left(2.2\right)$.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

As $n$ increases, then the Taylor polynomial approximation improves in size.

Suppose that the value being approximated is closer to the center of the series.

The approximation $p_3\left(2.1\right)\approx f\left(2.1\right)$ is written as follows.

$\left|p_3\left(2.1\right)-f\left(2.1\right)\right|$

Also the approximation $p_2\left(2.2\right)\approx f\left(2.2\right)$ is written as $\left|p_2\left(2.2\right)-f\left(2.2\right)\right|$.

Note that the value 2.1 is closer with the value 2 when compared with the value 2.2.

Also note that the number $3>2$.

Then the above approximation becomes,

$\left|p_3\left(2.1\right)-f\left(2.1\right)\right|<\left|p_3\left(2.1\right)-f\left(2.1\right)\right|$.

Therefore, the statement is true.

To determine

Whether the statement “If the Taylor series for f centered at 3 has a radius of convergence of 6, then the interval of convergence is $\left[-3,9\right]$.” is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The given Taylor series f is centered at 3 and the radius of convergence of the Taylor series is 6.

Note that the interval of convergence may or may not include the end points.

Therefore, the interval of convergence of the Taylor series may or may not include the end points.

Thus, the statement is false.

To determine

Whether the statement “The interval of convergence of the power series ${\displaystyle \sum c_k{x}^k}$ could be $\left(-\frac{7}{3},\frac{7}{3}\right)$.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The given power series is ${\displaystyle \sum c_k{x}^k}$ and the radius of convergence of the power series is centered at 0.

Note that the interval of convergence may or may note includes the end points.

Therefore, the interval of convergence of the Taylor series may not include the end points.

Thus, the statement is true.

To determine

Whether the statement “The Maclaurin series for $f\left(x\right)={\left(1+x\right)}^{12}$ has a finite number of nonzero terms.” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The given Maclaurin series is $f\left(x\right)={\left(1+x\right)}^{12}$.

Note that the last nonzero derivative of f is $f^{12}\left(x\right)$.

Therefore, all the derivative of $f\left(x\right)={\left(1+x\right)}^{12}$ vanishes after a certain point.

Therefore, the statement is true.

To determine

Whether the statement “If the power series ${\displaystyle \sum c_k{\left(x-3\right)}^k}$ has a radius of convergence of $R=4$ and converges at the endpoints of its interval of convergence, then its interval of convergence is $\left[-1,7\right]$” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

The given Power series is ${\displaystyle \sum c_k{\left(x-3\right)}^k}$.

Note that the series ${\displaystyle \sum c_k{\left(x-3\right)}^k}$ converges as $\left|r\right|<4$.

$\begin{array}{l}\left|x-3\right|<4\\ -4<x-3<4\\ -1<x<7\end{array}$

Note that, the interval of convergence is $\left[-1,7\right]$.

Therefore, the statement is true.