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sin((pi/180)*(4n-1))

Suma de la serie sin((pi/180)*(4n-1))



=

Solución

Ha introducido [src]
  45                    
 ___                    
 \  `                   
  \      / pi          \
   )  sin|---*(4*n - 1)|
  /      \180          /
 /__,                   
n = 1                   
$$\sum_{n=1}^{45} \sin{\left(\frac{\pi}{180} \left(4 n - 1\right) \right)}$$
Sum(sin((pi/180)*(4*n - 1)), (n, 1, 45))
Velocidad de la convergencia de la serie
Respuesta [src]
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\/ 2    \/ 6      ___    /  5   \/ 5       ___ |1   \/ 5 |   |\/    8     8            \4     4  /| |      \/    8     8            \4     4  /|   |\/    8     8            \4     4  /| |        \/    8     8            \4     4  /|   |\/    8     8            \4     4  /| |        \/    8     8            \4     4  /|   |\/    8     8            \4     4  /| |      \/    8     8            \4     4  /|   |  \/    8     8            \4     4  /| |      \/    8     8            \4     4  /|   |  \/    8     8            \4     4  /| |      \/    8     8            \4     4  /|   |  \/    8     8            \4     4  /| |      \4     4  /         \/    8     8   |   |  \/    8     8            \4     4  /| |      \4     4  /         \/    8     8   |   |      \/    8     8            \4     4  /| |1   \/ 5          \/    8     8   |   |      \/    8     8            \4     4  /| |  1   \/ 5          \/    8     8   |   |      \/    8     8            \4     4  /| |  1   \/ 5          \/    8     8   |   |      \/    8     8            \4     4  /| |1   \/ 5          \/    8     8   |   |      \4     4  /         \/    8     8   | |  1   \/ 5          \/    8     8   |   |      \4     4  /         \/    8     8   | |1   \/ 5          \/    8     8   |   |        \/    8     8            \4     4  /| |  1   \/ 5          \/    8     8   |   |        \/    8     8            \4     4  /| |1   \/ 5          \/    8     8   |      /pi\      / pi\      /5*pi\      /7*pi\      /7*pi\      /11*pi\      /11*pi\      /13*pi\      /13*pi\      /17*pi\      /17*pi\      /19*pi\      /23*pi\      /29*pi\      /31*pi\      /37*pi\      /41*pi\      /43*pi\      /47*pi\      /49*pi\      /53*pi\      /59*pi\      /61*pi\      /67*pi\      /71*pi\      /73*pi\      /77*pi\      /79*pi\      /83*pi\      /89*pi\
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  2       2           \/    8     8            \4     4  /   \       2                   2        / \          2                      2        /   \       2                   2        / \            2                      2        /   \       2                   2        / \            2                      2        /   \       2                   2        / \          2                      2        /   \         2                   2        / \          2                      2        /   \         2                   2        / \          2                      2        /   \         2                   2        / \        2                     2           /   \         2                   2        / \        2                     2           /   \          2                      2        / \8     8               2           /   \          2                      2        / \  8     8               2           /   \          2                      2        / \  8     8               2           /   \          2                      2        / \8     8               2           /   \        2                     2           / \  8     8               2           /   \        2                     2           / \8     8               2           /   \            2                      2        / \  8     8               2           /   \            2                      2        / \8     8               2           /      \36/      \180/      \ 36 /      \ 36 /      \180 /      \  36 /      \ 180 /      \  36 /      \ 180 /      \  36 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /      \ 180 /
$$\left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) \left(- \frac{\sqrt{5}}{8} + \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + \left(- \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} - \frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) \left(- \frac{\sqrt{5}}{8} - \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) \left(\frac{\sqrt{3} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + \left(- \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{3} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}\right) + \left(- \frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}\right) \left(- \frac{\sqrt{5}}{8} - \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \sin{\left(\frac{\pi}{180} \right)} + \sin{\left(\frac{\pi}{36} \right)} + \sin{\left(\frac{7 \pi}{180} \right)} + \left(- \frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}\right) \left(\frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \left(- \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) \left(- \frac{\sqrt{3} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) \left(- \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{3} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) + \sin{\left(\frac{11 \pi}{180} \right)} + \sin{\left(\frac{13 \pi}{180} \right)} + \sin{\left(\frac{17 \pi}{180} \right)} + \sin{\left(\frac{19 \pi}{180} \right)} + \left(\frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) \left(- \frac{\sqrt{3} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + \sin{\left(\frac{23 \pi}{180} \right)} + \left(- \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} - \frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) \left(- \frac{\sqrt{3} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \sin{\left(\frac{5 \pi}{36} \right)} + \left(\frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) \left(- \frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + \sin{\left(\frac{29 \pi}{180} \right)} + \sin{\left(\frac{31 \pi}{180} \right)} + \sin{\left(\frac{7 \pi}{36} \right)} + \left(- \frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) \left(- \frac{\sqrt{5}}{8} + \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + \sin{\left(\frac{37 \pi}{180} \right)} + \sin{\left(\frac{41 \pi}{180} \right)} + \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) \left(- \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{3} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) + \sin{\left(\frac{43 \pi}{180} \right)} + \frac{\sqrt{2}}{2} + \sin{\left(\frac{47 \pi}{180} \right)} + \sin{\left(\frac{49 \pi}{180} \right)} + \sin{\left(\frac{53 \pi}{180} \right)} + \sin{\left(\frac{11 \pi}{36} \right)} + \sin{\left(\frac{59 \pi}{180} \right)} + \left(- \frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) \left(- \frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + \sin{\left(\frac{61 \pi}{180} \right)} + \left(\frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}\right) \left(\frac{1}{8} + \frac{\sqrt{5}}{8} + \frac{\sqrt{3} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + \sin{\left(\frac{13 \pi}{36} \right)} + \sin{\left(\frac{67 \pi}{180} \right)} + \sin{\left(\frac{71 \pi}{180} \right)} + \sin{\left(\frac{73 \pi}{180} \right)} + \sin{\left(\frac{77 \pi}{180} \right)} + \sin{\left(\frac{79 \pi}{180} \right)} + \left(\frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}\right) \left(\frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{3} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2}\right) + \sin{\left(\frac{83 \pi}{180} \right)} + \sin{\left(\frac{17 \pi}{36} \right)} + \sin{\left(\frac{89 \pi}{180} \right)} + \sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right) + \frac{\sqrt{6}}{2} + \sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
sqrt(2)/2 + sqrt(6)/2 + sqrt(2)*sqrt(5/8 + sqrt(5)/8) + sqrt(2)*(1/4 + sqrt(5)/4) + (sqrt(5/8 - sqrt(5)/8)/2 + sqrt(3)*(1/4 + sqrt(5)/4)/2)*(sqrt(2)*sqrt(5/8 - sqrt(5)/8)/2 + sqrt(2)*(1/4 + sqrt(5)/4)/2) + (sqrt(5/8 - sqrt(5)/8)/2 - sqrt(3)*(1/4 + sqrt(5)/4)/2)*(-sqrt(2)*sqrt(5/8 - sqrt(5)/8)/2 - sqrt(2)*(1/4 + sqrt(5)/4)/2) + (sqrt(5/8 + sqrt(5)/8)/2 + sqrt(3)*(1/4 - sqrt(5)/4)/2)*(-sqrt(2)*sqrt(5/8 + sqrt(5)/8)/2 - sqrt(2)*(1/4 - sqrt(5)/4)/2) + (sqrt(5/8 + sqrt(5)/8)/2 - sqrt(3)*(1/4 - sqrt(5)/4)/2)*(sqrt(2)*sqrt(5/8 + sqrt(5)/8)/2 + sqrt(2)*(1/4 - sqrt(5)/4)/2) + (-sqrt(5/8 - sqrt(5)/8)/2 + sqrt(3)*(1/4 + sqrt(5)/4)/2)*(sqrt(2)*sqrt(5/8 - sqrt(5)/8)/2 - sqrt(2)*(1/4 + sqrt(5)/4)/2) + (-sqrt(5/8 - sqrt(5)/8)/2 - sqrt(3)*(1/4 + sqrt(5)/4)/2)*(sqrt(2)*sqrt(5/8 - sqrt(5)/8)/2 - sqrt(2)*(1/4 + sqrt(5)/4)/2) + (-sqrt(5/8 + sqrt(5)/8)/2 + sqrt(3)*(1/4 - sqrt(5)/4)/2)*(sqrt(2)*(1/4 - sqrt(5)/4)/2 - sqrt(2)*sqrt(5/8 + sqrt(5)/8)/2) + (-sqrt(5/8 + sqrt(5)/8)/2 - sqrt(3)*(1/4 - sqrt(5)/4)/2)*(sqrt(2)*(1/4 - sqrt(5)/4)/2 - sqrt(2)*sqrt(5/8 + sqrt(5)/8)/2) + (sqrt(2)*sqrt(5/8 - sqrt(5)/8)/2 + sqrt(2)*(1/4 + sqrt(5)/4)/2)*(1/8 + sqrt(5)/8 + sqrt(3)*sqrt(5/8 - sqrt(5)/8)/2) + (sqrt(2)*sqrt(5/8 + sqrt(5)/8)/2 + sqrt(2)*(1/4 - sqrt(5)/4)/2)*(-1/8 + sqrt(5)/8 + sqrt(3)*sqrt(5/8 + sqrt(5)/8)/2) + (sqrt(2)*sqrt(5/8 + sqrt(5)/8)/2 - sqrt(2)*(1/4 - sqrt(5)/4)/2)*(-1/8 + sqrt(5)/8 + sqrt(3)*sqrt(5/8 + sqrt(5)/8)/2) + (sqrt(2)*sqrt(5/8 + sqrt(5)/8)/2 - sqrt(2)*(1/4 - sqrt(5)/4)/2)*(1/8 - sqrt(5)/8 + sqrt(3)*sqrt(5/8 + sqrt(5)/8)/2) + (sqrt(2)*(1/4 + sqrt(5)/4)/2 - sqrt(2)*sqrt(5/8 - sqrt(5)/8)/2)*(-1/8 - sqrt(5)/8 + sqrt(3)*sqrt(5/8 - sqrt(5)/8)/2) + (sqrt(2)*(1/4 + sqrt(5)/4)/2 - sqrt(2)*sqrt(5/8 - sqrt(5)/8)/2)*(1/8 + sqrt(5)/8 + sqrt(3)*sqrt(5/8 - sqrt(5)/8)/2) + (-sqrt(2)*sqrt(5/8 - sqrt(5)/8)/2 - sqrt(2)*(1/4 + sqrt(5)/4)/2)*(-1/8 - sqrt(5)/8 + sqrt(3)*sqrt(5/8 - sqrt(5)/8)/2) + (-sqrt(2)*sqrt(5/8 + sqrt(5)/8)/2 - sqrt(2)*(1/4 - sqrt(5)/4)/2)*(1/8 - sqrt(5)/8 + sqrt(3)*sqrt(5/8 + sqrt(5)/8)/2) + sin(pi/36) + sin(pi/180) + sin(5*pi/36) + sin(7*pi/36) + sin(7*pi/180) + sin(11*pi/36) + sin(11*pi/180) + sin(13*pi/36) + sin(13*pi/180) + sin(17*pi/36) + sin(17*pi/180) + sin(19*pi/180) + sin(23*pi/180) + sin(29*pi/180) + sin(31*pi/180) + sin(37*pi/180) + sin(41*pi/180) + sin(43*pi/180) + sin(47*pi/180) + sin(49*pi/180) + sin(53*pi/180) + sin(59*pi/180) + sin(61*pi/180) + sin(67*pi/180) + sin(71*pi/180) + sin(73*pi/180) + sin(77*pi/180) + sin(79*pi/180) + sin(83*pi/180) + sin(89*pi/180)
Respuesta numérica [src]
28.6493442492750917383063418676
28.6493442492750917383063418676
Gráfico
Suma de la serie sin((pi/180)*(4n-1))

    Ejemplos de hallazgo de la suma de la serie