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Suma de la serie (n*x^(n-1)-10x^n+n*x^n)/((1-x)^n)



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Solución

Ha introducido [src]
  10                         
____                         
\   `                        
 \       n - 1       n      n
  \   n*x      - 10*x  + n*x 
   )  -----------------------
  /                  n       
 /            (1 - x)        
/___,                        
n = 2                        
$$\sum_{n=2}^{10} \frac{n x^{n} + \left(n x^{n - 1} - 10 x^{n}\right)}{\left(1 - x\right)^{n}}$$
Sum((n*x^(n - 1) - 10*x^n + n*x^n)/(1 - x)^n, (n, 2, 10))
Respuesta [src]
   9      8        8      7        7      6        6      5        5      4        4      3        3      2        2               9  
- x  + 9*x    - 2*x  + 8*x    - 3*x  + 7*x    - 4*x  + 6*x    - 5*x  + 5*x    - 6*x  + 4*x    - 7*x  + 3*x    - 8*x  + 2*x     10*x   
----------- + ------------- + ------------- + ------------- + ------------- + ------------- + ------------- + ------------ + ---------
         9              8               7               6               5               4               3              2            10
  (1 - x)        (1 - x)         (1 - x)         (1 - x)         (1 - x)         (1 - x)         (1 - x)        (1 - x)      (1 - x)  
$$\frac{10 x^{9}}{\left(1 - x\right)^{10}} + \frac{- 8 x^{2} + 2 x}{\left(1 - x\right)^{2}} + \frac{- 7 x^{3} + 3 x^{2}}{\left(1 - x\right)^{3}} + \frac{- 6 x^{4} + 4 x^{3}}{\left(1 - x\right)^{4}} + \frac{- 5 x^{5} + 5 x^{4}}{\left(1 - x\right)^{5}} + \frac{- 4 x^{6} + 6 x^{5}}{\left(1 - x\right)^{6}} + \frac{- 3 x^{7} + 7 x^{6}}{\left(1 - x\right)^{7}} + \frac{- 2 x^{8} + 8 x^{7}}{\left(1 - x\right)^{8}} + \frac{- x^{9} + 9 x^{8}}{\left(1 - x\right)^{9}}$$
(-x^9 + 9*x^8)/(1 - x)^9 + (-2*x^8 + 8*x^7)/(1 - x)^8 + (-3*x^7 + 7*x^6)/(1 - x)^7 + (-4*x^6 + 6*x^5)/(1 - x)^6 + (-5*x^5 + 5*x^4)/(1 - x)^5 + (-6*x^4 + 4*x^3)/(1 - x)^4 + (-7*x^3 + 3*x^2)/(1 - x)^3 + (-8*x^2 + 2*x)/(1 - x)^2 + 10*x^9/(1 - x)^10

    Ejemplos de hallazgo de la suma de la serie