Respuestas - Capítulo 5

Aquí encontrarás las respuestas a todos los ejercicios del capítulo 5

  1. f(x)=94+∑n=1∞5n2π2[(−1)n−1]cosnπ5x+5nπ(−1)n+1sennπ5x\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. f(x)=32+∑n=1∞2n2π2[1−(−1)n]cosnπ2x+2nπ(−1)n+1sennπ2x\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. (x)=12π(eπ−π−1)+∑n=1∞eπ(−1)n−1π(1+n2)cos(nx)+1π(neπ(−1)n+11+n2+n1+n2+(−1)nn−1n)sen(nx)\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. f(x)=3+∑n=1∞4n(−1)nsen(nx)\hspace{0.5cm} f(x)=3+\sum_{n=1}^{\infty}{\dfrac{4}{n}\left(-1\right)^nsen(nx)}

  5. f(x)=12π(eπ−e−π)+∑n=1∞(−1)n(eπ−e−π)π(1+n2)cos(nx)+(−1)nn(e−π−eπ)π(1+n2)sen(nx)\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. f(x)=38+∑n=1∞2n2π2(cosnπ2−1)cosnπ2x+2n2π2(sennπ2+nπ2(−1)n+1)sennπ2x\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. f(x)=−14+∑n=1∞−1nπsennπ2cosnπ2x+3nπ(1−cosnπ2)sennπ2x\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. f(x)=∑n=1∞−2nπsen(nπx)\hspace{0.5cm} f(x)=\sum_{n=1}^{\infty}{\dfrac{-2}{n\pi}sen(n\pi x)}

  9. f(x)=−14+∑n=1∞−1nπsennπ2cosnπ2x+3nπ(1−cosnπ2)sennπ2x\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. f(x)=23π2+∑n=1∞4n2(−1)n+1cos(nx)\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. f(x)=∑n=1∞[2π2n(−1)n+1+12n3(−1)n]sen(nx)\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. f(x)=2π+∑n=1∞4(−1)n+1π4n2−1cos(2nx)\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. f(x)=∑n=1∞[4n2πsennπ2+2n(−1)n+1]senn2x\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. f(x)=34+∑n=1∞4n2π2(cosnπ2−1)cosnπ2x\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}