A.11.2: Section 11.2 Answers

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2. \(F(x)=2+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\sin n\pi x;\quad F(x)=\left\{\begin{array}{cl}{2,}&{x=-1,}\\{2-x,}&{-1<x<1,}\\{2,}&{x=1}\end{array} \right.\)

3. \(F(x)=-\pi ^{2}-12\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}\cos nx-4\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\sin nx;\quad F(x)=\left\{\begin{array}{cl}{-3\pi ^{2},}&{x=-\pi ,}\\{2x-3x^{2},}&{-\pi <x<\pi ,}\\{3\pi ^{2},}&{x=\pi }\end{array} \right.\)

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. \(F(x)=\frac{1}{2}\sin\pi x-\frac{4}{\pi}\sum_{n=1}^{\infty}(-1)^{n}\frac{n}{4n^{2}-1}\sin 2n\pi x;\quad F(x)=\left\{\begin{array}{cl}{0,}&{-1\leq x<\frac{1}{2},}\\{-\frac{1}{2},}&{x=-\frac{1}{2},}\\{\sin\pi x,}&{-\frac{1}{2}<x<\frac{1}{2},}\\{\frac{1}{2},}&{x=\frac{1}{2},}\\{0,}&{\frac{1}{2}<x\leq 1}\end{array} \right.\)

13. \(F(x)=\frac{1}{\pi}+\frac{1}{\pi}\cos\pi x-\frac{2}{\pi}\sum_{n=2}^{\infty}\frac{1}{n^{2}-1}\left(1-n\sin\frac{n\pi}{2}\right)\cos n\pi x;\quad F(x)=\left\{\begin{array}{cl}{0,}&{-1\leq x<\frac{1}{2},}\\{\frac{1}{2},}&{x=-1,}\\{|\sin\pi x|,}&{-\frac{1}{2}<x<\frac{1}{2},}\\{\frac{1}{2},}&{x=1,}\\{0,}&{\frac{1}{2}<x\leq 1} \end{array}\right.\)

14. \(F(x)=\frac{1}{\pi ^{2}}+\frac{1}{4\pi}\cos\pi x+\frac{2}{\pi ^{2}}\sum_{n=1}^{\infty}(-1)^{n}\frac{4n^{2}+1}{(4n^{2}-1)^{2}}\cos2n\pi x+\frac{1}{4\pi}\sum_{n=1}^{\infty}(-1)^{n}\frac{2n+1}{n(n+1)}\cos (2n+1)\pi x;\quad F(x)=\left\{\begin{array}{cl}{0,}&{-1\leq x<\frac{1}{2},}\\{\frac{1}{4},}&{x=-\frac{1}{2},}\\{x\sin\pi x,}&{-\frac{1}{2}<x<\frac{1}{2},}\\{\frac{1}{4},}&{x=\frac{1}{2},}\\{0,}&{\frac{1}{2}<x\leq 1,}\end{array} \right.\)

15. \(F(x)=1-\frac{8}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{1}{(2n+1)^{2}}\cos\frac{(2n+1)\pi x}{4}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\sin\frac{n\pi x}{4};\quad F(x)=\left\{\begin{array}{cl}{2,}&{x=-4,}\\{0,}&{-4<x<0,}\\{x,}&{0\leq x<4,}\\{2,}&{x=4}\end{array} \right.\)

16. \(F(x)=\frac{1}{2}+\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\sin 2n\pi x+\frac{8}{\pi ^{3}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{3}}\sin(2n+1)\pi x;\quad F(x)=\left\{\begin{array}{cl}{\frac{1}{2},}&{x=-1,}\\{x^{2},}&{-1<x<0,}\\{\frac{1}{2},}&{x=0,}\\{1-x^{2},}&{0<x<1,}\\{\frac{1}{2},}&{x=1}\end{array} \right.\)

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. \(F(x)=\frac{\sin k\pi}{\pi}\left[\frac{1}{k}-2k\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}-k^{2}}\cos nx\right]\)