11.13: Additional Exercises
- Page ID
- 546
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)These problems require the techniques of this chapter, and are in no particular order. Some problems may be done in more than one way.
Exercises 11.12
Determine whether the series converges.
Ex 11.12.1 \(\sum_{n=0}^\infty {n\over n^2+4}\) (answer)
Ex 11.12.2 \( {1\over 1\cdot 2}+{1\over 3\cdot 4}+{1\over 5\cdot 6}+{1\over 7\cdot 8}+\cdots\) (answer)
Ex 11.12.3 \(\sum_{n=0}^\infty {n\over (n^2+4)^2}\) (answer)
Ex 11.12.4 \(\sum_{n=0}^\infty {n!\over 8^n}\) (answer)
Ex 11.12.5 \(1-{3\over4}+{5\over8}-{7\over12}+{9\over16}+\cdots\) (answer)
Ex 11.12.6 \(\sum_{n=0}^\infty {1\over \sqrt{n^2+4}}\) (answer)
Ex 11.12.7 \(\sum_{n=0}^\infty {\sin^3(n)\over n^2}\) (answer)
Ex 11.12.8 \(\sum_{n=0}^{\infty} {n\over e^n}\) (answer)
Ex 11.12.9 \(\sum_{n=0}^\infty {n!\over 1\cdot3\cdot5\cdots(2n-1)}\) (answer)
Ex 11.12.10 \(\sum_{n=1}^\infty {1\over n\sqrt n}\) (answer)
Ex 11.12.11 \({1\over 2\cdot 3\cdot 4}+{2\over 3\cdot 4\cdot 5}+{3\over 4\cdot 5\cdot 6}+{4\over 5\cdot 6 \cdot 7}+\cdots\) (answer)
Ex 11.12.12 \(\sum_{n=1}^\infty {1\cdot3\cdot5\cdots(2n-1)\over (2n)!}\) (answer)
Ex 11.12.13 \(\sum_{n=0}^\infty {6^n\over n!}\) (answer)
Ex 11.12.14 \(\sum_{n=1}^\infty {(-1)^{n-1}\over\sqrt n}\) (answer)
Ex 11.12.15 \(\sum_{n=1}^\infty {2^n 3^{n-1}\over n!}\) (answer)
Ex 11.12.16 \(1+ {5^2\over 2^2}+{5^4\over (2\cdot4)^2} +{5^6\over(2\cdot4\cdot6)^2}+ {5^8\over(2\cdot4\cdot6\cdot8)^2}+\cdots\) (answer)
Ex 11.12.17 \(\sum_{n=1}^\infty \sin(1/n)\) (answer)
Find the interval and radius of convergence; you need not check the endpoints of the intervals.
Ex 11.12.18 \(\sum_{n=0}^\infty {2^n\over n!}x^n\) (answer)
Ex 11.12.19 \(\sum_{n=0}^\infty {x^n\over 1+3^n}\) (answer)
Ex 11.12.20 \(\sum_{n=1}^\infty {x^n\over n3^n}\) (answer)
Ex 11.12.21 \( x+{1\over 2}{x^3\over3} + {1\cdot 3\over 2\cdot4}{x^5\over5}+ {1\cdot 3\cdot5\over 2\cdot4\cdot6}{x^7\over7}+\cdots\) (answer)
Ex 11.12.22 \(\sum_{n=1}^\infty {n!\over n^2} x^n\) (answer)
Ex 11.12.2 \(\sum_{n=1}^\infty {(-1)^n\over n^2 3^n} x^{2n}\) (answer)
Ex 11.12.24 \(\sum_{n=0}^\infty {(x-1)^n\over n!}\) (answer)
Find a series for each function, using the formula for Maclaurin series and algebraic manipulation as appropriate.
Ex 11.12.25 \( 2^x\) (answer)
Ex 11.12.26 \(\ln(1+x)\) (answer)
Ex 11.12.27 \( \ln\left({1+x\over 1-x}\right)\) (answer)
Ex 11.12.28 \(\sqrt{1+x}\) (answer)
Ex 11.12.29 \({1\over 1+x^2}\) (answer)
Ex 11.12.30 \(\arctan(x)\) (answer)
Ex 11.12.31Use the answer to the previous problem to discover a series for a well-known mathematical constant. (answer)