10.R: Chapter 10 Review Exercises
- Page ID
- 72444
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)True or False? In exercises 1 - 4, 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 \(5\), then the radius of convergence for the series \(\displaystyle \sum_{n=1}^∞na_nx^{n−1}\) is also \(5\).
- Answer
- True
2) Power series can be used to show that the derivative of \(e^x\) is \(e^x\). (Hint: Recall that \(\displaystyle e^x=\sum_{n=0}^∞\frac{1}{n!}x^n.\))
3) For small values of \(x,\) \(\sin x ≈ x.\)
- Answer
- True
4) The radius of convergence for the Maclaurin series of \(f(x)=3^x\) is \(3\).
In exercises 5 - 8, find the radius of convergence and the interval of convergence for the given series.
5) \(\displaystyle \sum_{n=0}^∞n^2(x−1)^n\)
- Answer
- ROC: \(1\); IOC: \((0,2)\)
6) \(\displaystyle \sum_{n=0}^∞\frac{x^n}{n^n}\)
7) \(\displaystyle \sum_{n=0}^∞\frac{3nx^n}{12^n}\)
- Answer
- ROC: \(12;\) IOC: \((−16,8)\)
8) \(\displaystyle \sum_{n=0}^∞\frac{2^n}{e^n}(x−e)^n\)
In exercises 9 - 10, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.
9) \(f(x)=\dfrac{x^2}{x+3}\)
- Answer
- \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{3^{n+1}}x^n;\) ROC: \(3\); IOC: \((−3,3)\)
10) \(f(x)=\dfrac{8x+2}{2x^2−3x+1}\)
In exercises 11 - 12, find the power series for the given function using term-by-term differentiation or integration.
11) \(f(x)=\tan^{−1}(2x)\)
- Answer
- integration: \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{2n+1}(2x)^{2n+1}\)
12) \(f(x)=\dfrac{x}{(2+x^2)^2}\)
In exercises 13 - 14, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?
13) \(f(x)=x^3−2x^2+4, \quad a=−3\)
- Answer
- \(p_4(x)=(x+3)^3−11(x+3)^2+39(x+3)−41;\) exact
14) \(f(x)=e^{1/(4x)}, \quad a=4\)
In exercises 15 - 16, find the Maclaurin series for the given function.
15) \(f(x)=\cos(3x)\)
- Answer
- \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n(3x)^{2n}}{2n!}\)
16) \(f(x)=\ln(x+1)\)
In exercises 17 - 18, find the Taylor series at the given value.
17) \(f(x)=\sin x, \quad a=\frac{π}{2}\)
- Answer
- \(\displaystyle \sum_{n=0}^∞\frac{(−1)^n}{(2n)!}\left(x−\frac{π}{2}\right)^{2n}\)
18) \(f(x)=\dfrac{3}{x},\quad a=1\)
In exercises 19 - 20, find the Maclaurin series for the given function.
19) \(f(x)=e^{−x^2}−1\)
- Answer
- \(\displaystyle \sum_{n=1}^∞\frac{(−1)^n}{n!}x^{2n}\)
20) \(f(x)=\cos x−x\sin x\)
In exercises 21 - 23, find the Maclaurin series for \(\displaystyle F(x)=∫^x_0f(t)\,dt\) by integrating the Maclaurin series of \(f(x)\) term by term.
21) \(f(x)=\dfrac{\sin x}{x}\)
- Answer
- \(\displaystyle F(x)=\sum_{n=0}^∞\frac{(−1)^n}{(2n+1)(2n+1)!}x^{2n+1}\)
22) \(f(x)=1−e^x\)
23) Use power series to prove Euler’s formula: \(e^{ix}=cosx+isinx\)
- Answer
- Answers may vary.
Exercises 24 - 26 consider problems of annuity payments.
24) For annuities with a present value of \($1\) million, calculate the annual payouts given over \(25\) years assuming interest rates of \(1\%,5\%\), and \(10\%.\)
25) A lottery winner has an annuity that has a present value of \($10\) million. What interest rate would they need to live on perpetual annual payments of \($250,000\)?
- Answer
- \(2.5\%\)
26) Calculate the necessary present value of an annuity in order to support annual payouts of \($15,000\) given over \(25\) years assuming interest rates of \(1\%,5\%\),and \(10\%.\)