1 E: Chapter Exercises

Exercise $\PageIndex{1}$

True or False? In the following exercises, justify your answer with a proof or a counterexample.

1. If the radius of convergence for a power series $\displaystyle \sum_{n=0}^∞a_nx^n$ is $\displaystyle 5$, then the radius of convergence for the series $\displaystyle \sum_{n=1}^∞na_nx^{n−1}$ is also $\displaystyle 5$.

Answer

True

2. Power series can be used to show that the derivative of $\displaystyle e^x$ is $\displaystyle e^x$. (Hint: Recall that $\displaystyle e^x=\sum_{n=0}^∞\frac{1}{n!}x^n.$)

3. For small values of $\displaystyle x,sinx≈x.$

Answer

True

4. The radius of convergence for the Maclaurin series of $\displaystyle f(x)=3^x$ is $\displaystyle 3$

Exercise $\PageIndex{2}$

In the following exercises, find the radius of convergence and the interval of convergence for the given series.

1. $\displaystyle \sum_{n=0}^∞n^2(x−1)^n$

Answer

ROC: $\displaystyle 1$; IOC: $\displaystyle (0,2)$

2. $\displaystyle \sum_{n=0}^∞\frac{x^n}{n^n}$

3. $\displaystyle \sum_{n=0}^∞\frac{3nx^n}{12^n}$

Answer

ROC: $\displaystyle 12;$ IOC: $\displaystyle (−16,8)$

4. $\displaystyle \sum_{n=0}^∞\frac{2^n}{e^n}(x−e)^n$

Exercise $\PageIndex{3}$

In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

1. $\displaystyle f(x)=\frac{x^2}{x+3}$

Answer

$\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{3^{n+1}}x^n;$ ROC: $\displaystyle 3$; IOC: $\displaystyle (−3,3)$

2. $\displaystyle f(x)=\frac{8x+2}{2x^2−3x+1}$

Exercise $\PageIndex{4}$

In the following exercises, find the power series for the given function using term-by-term differentiation or integration.

1. $\displaystyle f(x)=tan^{−1}(2x)$

Answer

integration: $\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{2n+1}(2x)^{2n+1}$

2. $\displaystyle f(x)=\frac{x}{(2+x^2)^2}$

Exercise $\PageIndex{5}$

In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

1. $\displaystyle f(x)=x^3−2x^2+4,a=−3$

Answer

$\displaystyle p_4(x)=(x+3)^3−11(x+3)^2+39(x+3)−41;$ exact

2. $\displaystyle f(x)=e^{1/(4x)},a=4$

Exercise $\PageIndex{6}$

In the following exercises, find the Maclaurin series for the given function.

1. $\displaystyle f(x)=cos(3x)$

Answer

$\displaystyle \sum_{n=0}^∞\frac{(−1)^n(3x)^{2n}}{2n!}$

2. $\displaystyle f(x)=ln(x+1)$

Exercise $\PageIndex{7}$

In the following exercises, find the Taylor series at the given value.

1. $\displaystyle f(x)=sinx,a=\frac{π}{2}$

Answer

$\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{(2n)!}(x−\frac{π}{2})^{2n}$

2. $\displaystyle f(x)=\frac{3}{x},a=1$

Exercise $\PageIndex{8}$

In the following exercises, find the Maclaurin series for the given function.

1. $\displaystyle f(x)=e^{−x^2}−1$

Answer

$\displaystyle \sum_{n=1}^∞\frac{(−1)^n}{n!}x^{2n}$

2. $\displaystyle f(x)=cosx−xsinx$

Exercise $\PageIndex{9}$

In the following exercises, find the Maclaurin series for $\displaystyle F(x)=∫^x_0f(t)dt$ by integrating the Maclaurin series of $\displaystyle f(x)$ term by term.

1. $\displaystyle f(x)=\frac{sinx}{x}$

Answer

$\displaystyle F(x)=\sum_{n=0}^∞\frac{(−1)^n}{(2n+1)(2n+1)!}x^{2n+1}$

2. $\displaystyle f(x)=1−e^x$

3. Use power series to prove Euler’s formula: $\displaystyle e^{ix}=cosx+isinx$

Exercise $\PageIndex{10}$

The following exercises consider problems of annuity payments.

1. For annuities with a present value of $\displaystyle $1$ million, calculate the annual payouts given over $\displaystyle 25$ years assuming interest rates of $\displaystyle 1%,5%$, and $\displaystyle 10%.$

2. A lottery winner has an annuity that has a present value of $\displaystyle $10$ million. What interest rate would they need to live on perpetual annual payments of $\displaystyle $250,000$?

Answer

$\displaystyle 2.5%$

3. Calculate the necessary present value of an annuity in order to support annual payouts of $\displaystyle $15,000$ given over $\displaystyle 25$ years assuming interest rates of $\displaystyle 1%,5%$,and $\displaystyle 10%.$