A.11.2: Section 11.2 Answers
- Page ID
- 43743
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2. \(F(x)=2+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\sin n\pi x;\quad F(x)=\left\{\begin{array}{cl}{2,}&{x=-1,}\\[4pt]{2-x,}&{-1<x<1,}\\[4pt]{2,}&{x=1}\end{array} \right.\)
3. \(F(x)=-\pi ^{2}-12\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}\cos nx-4\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\sin nx;\quad F(x)=\left\{\begin{array}{cl}{-3\pi ^{2},}&{x=-\pi ,}\\[4pt]{2x-3x^{2},}&{-\pi <x<\pi ,}\\[4pt]{3\pi ^{2},}&{x=\pi }\end{array} \right.\)
4. \(F(x)=-\frac{12}{\pi ^{2}}\sum_{n=1}^{\infty}(-1)^{n}\frac{\cos n\pi x}{n^{2}};\quad F(x)=1-3x^{2},\quad -1\leq x\leq 1\)
5. \(F(x)=\frac{2}{\pi}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{1}{4n^{2}-1}\cos 2nx;\quad F(x)=|\sin x|,\quad -\pi\leq x\leq\pi \)
6. \(F(x)=-\frac{1}{2}\sin x+2\sum_{n=2}^{\infty}(-1)^{n}\frac{n}{n^{2}-1}\sin nx;\quad F(x)=x\cos x,\quad -\pi\leq x\leq\pi\)
7. \(F(x)=-\frac{2}{\pi}+\frac{\pi}{2}\cos x-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{4n^{2}+1}{(4n^{2}-1)^{2}}\cos 2nx;\quad F(x)=|x|\cos x,\quad -\pi\leq x\leq\pi \)
8. \(F(x)=1-\frac{1}{2}\cos x-2\sum_{n=2}^{\infty}\frac{(-1)^{n}}{n^{2}-1}\cos nx;\quad F(x)=x\sin x,\quad -\pi\leq x\leq\pi\)
9. \(F(x)=\frac{\pi}{2}\sin x-\frac{16}{\pi}\sum_{n=1}^{\infty}\frac{n}{(4n^{2}-1)^{2}}\sin 2nx;\quad F(x)=|x|\sin x,\quad -\pi\leq x\leq\pi\)
10. \(F(x)=\frac{1}{\pi}+\frac{1}{2}\cos\pi x-\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{4n^{2}-1}\cos 2n\pi x;\quad F(x)=f(x),\quad -1\leq x\leq 1\)
11. \(F(x)=\frac{1}{4\pi}\sin\pi x-\frac{8}{\pi ^{2}}\sum_{n=1}^{\infty}(-1)^{n}\frac{n}{(4n^{2}-1)^{2}}\sin 2n\pi x;\quad -\frac{1}{4\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n(n+1)}\sin (2n+1)\pi x\quad F(x)=f(x),\quad -1\leq x\leq 1\)
12. \(F(x)=\frac{1}{2}\sin\pi x-\frac{4}{\pi}\sum_{n=1}^{\infty}(-1)^{n}\frac{n}{4n^{2}-1}\sin 2n\pi x;\quad F(x)=\left\{\begin{array}{cl}{0,}&{-1\leq x<\frac{1}{2},}\\[4pt]{-\frac{1}{2},}&{x=-\frac{1}{2},}\\[4pt]{\sin\pi x,}&{-\frac{1}{2}<x<\frac{1}{2},}\\[4pt]{\frac{1}{2},}&{x=\frac{1}{2},}\\[4pt]{0,}&{\frac{1}{2}<x\leq 1}\end{array} \right.\)
13. \(F(x)=\frac{1}{\pi}+\frac{1}{\pi}\cos\pi x-\frac{2}{\pi}\sum_{n=2}^{\infty}\frac{1}{n^{2}-1}\left(1-n\sin\frac{n\pi}{2}\right)\cos n\pi x;\quad F(x)=\left\{\begin{array}{cl}{0,}&{-1\leq x<\frac{1}{2},}\\[4pt]{\frac{1}{2},}&{x=-1,}\\[4pt]{|\sin\pi x|,}&{-\frac{1}{2}<x<\frac{1}{2},}\\[4pt]{\frac{1}{2},}&{x=1,}\\[4pt]{0,}&{\frac{1}{2}<x\leq 1} \end{array}\right.\)
14. \(F(x)=\frac{1}{\pi ^{2}}+\frac{1}{4\pi}\cos\pi x+\frac{2}{\pi ^{2}}\sum_{n=1}^{\infty}(-1)^{n}\frac{4n^{2}+1}{(4n^{2}-1)^{2}}\cos2n\pi x+\frac{1}{4\pi}\sum_{n=1}^{\infty}(-1)^{n}\frac{2n+1}{n(n+1)}\cos (2n+1)\pi x;\quad F(x)=\left\{\begin{array}{cl}{0,}&{-1\leq x<\frac{1}{2},}\\[4pt]{\frac{1}{4},}&{x=-\frac{1}{2},}\\[4pt]{x\sin\pi x,}&{-\frac{1}{2}<x<\frac{1}{2},}\\[4pt]{\frac{1}{4},}&{x=\frac{1}{2},}\\[4pt]{0,}&{\frac{1}{2}<x\leq 1,}\end{array} \right.\)
15. \(F(x)=1-\frac{8}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{1}{(2n+1)^{2}}\cos\frac{(2n+1)\pi x}{4}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\sin\frac{n\pi x}{4};\quad F(x)=\left\{\begin{array}{cl}{2,}&{x=-4,}\\[4pt]{0,}&{-4<x<0,}\\[4pt]{x,}&{0\leq x<4,}\\[4pt]{2,}&{x=4}\end{array} \right.\)
16. \(F(x)=\frac{1}{2}+\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\sin 2n\pi x+\frac{8}{\pi ^{3}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{3}}\sin(2n+1)\pi x;\quad F(x)=\left\{\begin{array}{cl}{\frac{1}{2},}&{x=-1,}\\[4pt]{x^{2},}&{-1<x<0,}\\[4pt]{\frac{1}{2},}&{x=0,}\\[4pt]{1-x^{2},}&{0<x<1,}\\[4pt]{\frac{1}{2},}&{x=1}\end{array} \right.\)
17. \(F(x)=\frac{3}{4}+\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\sin\frac{n\pi}{2}\cos\frac{n\pi x}{2}+\frac{3}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\left(\cos n\pi -\cos\frac{n\pi }{2}\right)\sin\frac{n\pi x}{2}\)
18. \(F(x)=\frac{5}{2}+\frac{3}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\sin\frac{2n\pi}{3}\cos\frac{n\pi x}{3}+\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\left(\cos n\pi -\cos\frac{2n\pi }{3}\right)\sin\frac{n\pi x}{3}\)
20. \(F(x)=\frac{\sinh \pi}{\pi}\left(1+2\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}+1}\cos nx-2\sum_{n=1}^{\infty}\frac{(-1)^{n}n}{n^{2}+1}\sinh nx\right)\)
21. \(F(x)=-\pi\cos x-\frac{1}{2}\sin x+2\sum_{n=2}^{\infty}(-1)^{n}\frac{n}{n^{2}-1}\sin nx\)
22. \(F(x)=1-\frac{1}{2}\cos x-\pi\sin x-2\sum_{n=2}^{\infty}\frac{(-1)^{n}}{n^{2}-1}\cos nx\)
23. \(F(x)=-\frac{2\sin k\pi}{\pi}\sum_{n=1}^{\infty}(-1)^{n}\frac{n}{n^{2}-k^{2}}\sin nx\)
24. \(F(x)=\frac{\sin k\pi}{\pi}\left[\frac{1}{k}-2k\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}-k^{2}}\cos nx\right]\)