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

If the radius of convergence for a power series $\sum _{n=0}^{\infty}{a}_{n}{x}^{n}$ is $5,$ then the radius of convergence for the series $\sum _{n=1}^{\infty}n{a}_{n}{x}^{n-1}$ is also $5.$

Power series can be used to show that the derivative of ${e}^{x}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}{e}^{x}.$ (*Hint:* Recall that ${e}^{x}={\displaystyle \sum _{n=0}^{\infty}\frac{1}{n\text{!}}}{x}^{n}.)$

The radius of convergence for the Maclaurin series of $f\left(x\right)={3}^{x}$ is $3.$

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

$\sum _{n=0}^{\infty}\frac{{x}^{n}}{{n}^{n}}$

$\sum _{n=0}^{\infty}\frac{{2}^{n}}{{e}^{n}}}{\left(x-e\right)}^{n$

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.

$f\left(x\right)=\frac{8x+2}{2{x}^{2}-3x+1}$

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

$f\left(x\right)=\frac{x}{{\left(2+{x}^{2}\right)}^{2}}$

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?

$f\left(x\right)={e}^{1\text{/}\left(4x\right)},a=4$

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

$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(x+1\right)$

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

$f\left(x\right)=\frac{3}{x},a=1$

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

$f\left(x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x-x\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x$

In the following exercises, find the Maclaurin series for $F\left(x\right)={\displaystyle {\int}_{0}^{x}f\left(t\right)}dt$ by integrating the Maclaurin series of $f\left(x\right)$ term by term.

$f\left(x\right)=1-{e}^{x}$

Use power series to prove Euler’s formula: ${e}^{ix}=\text{cos}\phantom{\rule{0.1em}{0ex}}x+i\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x$

The following exercises consider problems of annuity payments.

For annuities with a present value of $\text{\$}1$ million, calculate the annual payouts given over $25$ years assuming interest rates of $1\text{\%},5\text{\%},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}10\text{\%}.$

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

Calculate the necessary present value of an annuity in order to support annual payouts of $\text{\$}\mathrm{15,000}$ given over $25$ years assuming interest rates of $1\text{\%},5\text{\%},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}10\text{\%}.$