Respuestas - Capítulo 5

Aquí encontrarás las respuestas a todos los ejercicios del capítulo 5

1. $\hspace{0.5cm} f(x)=\dfrac{9}{4}+\sum_{n=1}^{\infty}{\dfrac{5}{n^2\pi^2}\left[\left(-1\right)^n-1\right]cos{\dfrac{n\pi}{5}}x+\dfrac{5}{n\pi}\left(-1\right)^{n+1}sen\dfrac{n\pi}{5}x}$
2. $\hspace{0.5cm} f(x)=\dfrac{3}{2}+\sum_{n=1}^{\infty}{\dfrac{2}{n^2\pi^2}\left[1-\left(-1\right)^n\right]cos{\dfrac{n\pi}{2}}x+\dfrac{2}{n\pi}\left(-1\right)^{n+1}sen\dfrac{n\pi}{2}x}$
3. $\hspace{0.5cm} (x)=\dfrac{1}{2\pi}\left(e^\pi-\pi-1\right)+\sum_{n=1}^{\infty}{\dfrac{e^\pi\left(-1\right)^n-1}{\pi\left(1+n^2\right)}cos{(nx)}+\dfrac{1}{\pi}\left(\dfrac{ne^\pi\left(-1\right)^{n+1}}{1+n^2}+\dfrac{n}{1+n^2}+\dfrac{\left(-1\right)^n}{n}-\dfrac{1}{n}\right)sen(nx)}$
4. $\hspace{0.5cm} f(x)=3+\sum_{n=1}^{\infty}{\dfrac{4}{n}\left(-1\right)^nsen(nx)}$
5. $\hspace{0.5cm} f(x)=\dfrac{1}{2\pi}\left(e^\pi-e^{-\pi}\right)+\sum_{n=1}^{\infty}{\dfrac{\left(-1\right)^n\left(e^\pi-e^{-\pi}\right)}{\pi\left(1+n^2\right)}cos{(n}x)+\dfrac{\left(-1\right)^nn\left(e^{-\pi}-e^\pi\right)}{\pi\left(1+n^2\right)}sen(nx)}$
6. $\hspace{0.5cm}f(x)=\dfrac{3}{8}+\sum_{n=1}^{\infty}{\dfrac{2}{n^2\pi^2}\left(cos{\dfrac{n\pi}{2}}-1\right)cos{\dfrac{n\pi}{2}}x+\dfrac{2}{n^2\pi^2}\left(sen\dfrac{n\pi}{2}+\dfrac{n\pi}{2}\left(-1\right)^{n+1}\right)sen\dfrac{n\pi}{2}x}$
7. $\hspace{0.5cm}f(x)=-\dfrac{1}{4}+\sum_{n=1}^{\infty}{\dfrac{-1}{n\pi}sen\dfrac{n\pi}{2}cos{\dfrac{n\pi}{2}}x+\dfrac{3}{n\pi}\left(1-cos{\dfrac{n\pi}{2}}\right)sen\dfrac{n\pi}{2}x}$
8. $\hspace{0.5cm} f(x)=\sum_{n=1}^{\infty}{\dfrac{-2}{n\pi}sen(n\pi x)}$
9. $\hspace{0.5cm} f(x)=-\dfrac{1}{4}+\sum_{n=1}^{\infty}{\dfrac{-1}{n\pi}sen\dfrac{n\pi}{2}cos{\dfrac{n\pi}{2}}x+\dfrac{3}{n\pi}\left(1-cos{\dfrac{n\pi}{2}}\right)sen\dfrac{n\pi}{2}x}$
10. $\hspace{0.5cm}f(x)=\dfrac{2}{3}\pi^2+\sum_{n=1}^{\infty}{\frac{4}{n^2}\left(-1\right)^{n+1}cos{(n}x)}$
11. $\hspace{0.5cm} f(x)=\sum_{n=1}^{\infty}{\left[\dfrac{2\pi^2}{n}\left(-1\right)^{n+1}+\dfrac{12}{n^3}\left(-1\right)^n\right]sen(nx)}$
12. $\hspace{0.5cm} f(x)=\dfrac{2}{\pi}+\sum_{n=1}^{\infty}{\dfrac{4\left(-1\right)^{n+1}}{\pi4n^2-1}cos{(2}nx)}$
13. $\hspace{0.5cm} f(x)=\sum_{n=1}^{\infty}\left[\dfrac{4}{n^2\pi}sen\dfrac{n\pi}{2}+\dfrac{2}{n}\left(-1\right)^{n+1}\right]sen\dfrac{n}{2}x$
14. $\hspace{0.5cm} f(x)=\dfrac{3}{4}+\sum_{n=1}^{\infty}{\dfrac{4}{n^2\pi^2}\left(cos{\dfrac{n\pi}{2}}-1\right)cos{\dfrac{n\pi}{2}}x}$