Sr Examen

Otras calculadoras

factorial(m)*b^(-10)*((b-1)/b)^(m-10)*5^m/(((factorial(10)*factorial(m-10)))*(1+5)^(m+1))
Suma de la serie/ factorial(m)*b^(-10)*((b-1)/b)^(m-10)*5^m/(((factorial(10)*factorial(m-10)))*(1+5)^(m+1))

Suma de la serie factorial(m)*b^(-10)*((b-1)/b)^(m-10)*5^m/(((factorial(10)*factorial(m-10)))*(1+5)^(m+1))


∑
=

Solución

Ha introducido [src] 
   oo                       
______                      
\     `                     
 \                 m - 10   
  \      m! /b - 1\        m
   \    ---*|-----|      *5 
    \    10 \  b  /         
    /   b                   
   /    --------------------
  /                    m + 1
 /      10!*(m - 10)!*6     
/_____,                     
 m = 0
$$\sum_{m=0}^{\infty} \frac{5^{m} \frac{m!}{b^{10}} \left(\frac{b - 1}{b}\right)^{m - 10}}{6^{m + 1} \cdot 10! \left(m - 10\right)!}$$ 
Sum((((factorial(m)/b^10)*((b - 1)/b)^(m - 10))*5^m)/(((factorial(10)*factorial(m - 10))*6^(m + 1))), (m, 0, oo))
Velocidad de la convergencia de la serie
Respuesta [src] 
/                     10                      
|              /    1\            |-1 + b|    
|   1708984375*|1 - -|          5*|------|    
|              \    b/            |  b   |    
| -----------------------   for ---------- < 1
|                      11           6         
|      /    5*(-1 + b)\                       
| 2916*|1 - ----------|                       
|      \       6*b    /                       
|                                             
<  oo                                         
|_____                                        
|\    `                                       
| \                   m                       
|  \     m  -m /    1\                        
|   \   5 *6  *|1 - -| *m!                    
|   /          \    b/          otherwise     
|  /    ------------------                    
| /         (-10 + m)!                        
|/____,                                       
\m = 0                                        
----------------------------------------------
                              10              
             21772800*(-1 + b)
$$\frac{\begin{cases} \frac{1708984375 \left(1 - \frac{1}{b}\right)^{10}}{2916 \left(1 - \frac{5 \left(b - 1\right)}{6 b}\right)^{11}} & \text{for}\: \frac{5 \left|{\frac{b - 1}{b}}\right|}{6} < 1 \\\sum_{m=0}^{\infty} \frac{5^{m} 6^{- m} \left(1 - \frac{1}{b}\right)^{m} m!}{\left(m - 10\right)!} & \text{otherwise} \end{cases}}{21772800 \left(b - 1\right)^{10}}$$ 
Piecewise((1708984375*(1 - 1/b)^10/(2916*(1 - 5*(-1 + b)/(6*b))^11), 5*Abs((-1 + b)/b)/6 < 1), (Sum(5^m*6^(-m)*(1 - 1/b)^m*factorial(m)/factorial(-10 + m), (m, 0, oo)), True))/(21772800*(-1 + b)^10)
Gráfico
Suma de la serie factorial(m)*b^(-10)*((b-1)/b)^(m-10)*5^m/(((factorial(10)*factorial(m-10)))*(1+5)^(m+1))

    Ejemplos de hallazgo de la suma de la serie

    © Sr Examen