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Suma de la serie sin^n(x)/(n+1)



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Solución

Ha introducido [src]
  oo         
____         
\   `        
 \       n   
  \   sin (x)
  /   -------
 /     n + 1 
/___,        
n = 1        
n=1sinn(x)n+1\sum_{n=1}^{\infty} \frac{\sin^{n}{\left(x \right)}}{n + 1}
Sum(sin(x)^n/(n + 1), (n, 1, oo))
Respuesta [src]
//    2      2*log(1 - sin(x))\                                            
||- ------ - -----------------|*sin(x)                                     
||  sin(x)           2        |                                            
|\                sin (x)     /                                            
|-------------------------------------  for And(|sin(x)| <= 1, sin(x) != 1)
|                  2                                                       
|                                                                          
|              oo                                                          
<            ____                                                          
|            \   `                                                         
|             \       n                                                    
|              \   sin (x)                                                 
|              /   -------                           otherwise             
|             /     1 + n                                                  
|            /___,                                                         
|            n = 1                                                         
\                                                                          
{(2log(1sin(x))sin2(x)2sin(x))sin(x)2forsin(x)1sin(x)1n=1sinn(x)n+1otherwise\begin{cases} \frac{\left(- \frac{2 \log{\left(1 - \sin{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)}} - \frac{2}{\sin{\left(x \right)}}\right) \sin{\left(x \right)}}{2} & \text{for}\: \left|{\sin{\left(x \right)}}\right| \leq 1 \wedge \sin{\left(x \right)} \neq 1 \\\sum_{n=1}^{\infty} \frac{\sin^{n}{\left(x \right)}}{n + 1} & \text{otherwise} \end{cases}
Piecewise(((-2/sin(x) - 2*log(1 - sin(x))/sin(x)^2)*sin(x)/2, (Ne(sin(x), 1))∧(Abs(sin(x)) <= 1)), (Sum(sin(x)^n/(1 + n), (n, 1, oo)), True))

    Ejemplos de hallazgo de la suma de la serie