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Suma de la serie (x-2)^(n+1)/3^n(n+2)



=

Solución

Ha introducido [src]
  oo                      
____                      
\   `                     
 \           n + 1        
  \   (x - 2)             
   )  ------------*(n + 2)
  /         n             
 /         3              
/___,                     
n = 1                     
$$\sum_{n=1}^{\infty} \frac{\left(x - 2\right)^{n + 1}}{3^{n}} \left(n + 2\right)$$
Sum(((x - 2)^(n + 1)/3^n)*(n + 2), (n, 1, oo))
Respuesta [src]
                                                  //        2   x                           \     //        2   x                           \                                                
    //        2   x                         \     ||      - - + -                           |     ||      - - + -                           |       //        2   x                         \
    ||      - - + -                         |     ||        3   3              |  2   x|    |     ||        3   3              |  2   x|    |       ||      - - + -                         |
    ||        3   3            |  2   x|    |     ||      --------         for |- - + -| < 1|     ||      --------         for |- - + -| < 1|       ||        3   3            |  2   x|    |
    ||      -------        for |- - + -| < 1|     ||             2             |  3   3|    |     ||             2             |  3   3|    |       ||      -------        for |- - + -| < 1|
    ||       5   x             |  3   3|    |     ||      /5   x\                           |     ||      /5   x\                           |       ||       5   x             |  3   3|    |
    ||       - - -                          |     ||      |- - -|                           |     ||      |- - -|                           |       ||       - - -                          |
    ||       3   3                          |     ||      \3   3/                           |     ||      \3   3/                           |       ||       3   3                          |
- 4*|<                                      | - 2*|<                                        | + x*|<                                        | + 2*x*|<                                      |
    ||  oo                                  |     ||  oo                                    |     ||  oo                                    |       ||  oo                                  |
    || ___                                  |     || ___                                    |     || ___                                    |       || ___                                  |
    || \  `                                 |     || \  `                                   |     || \  `                                   |       || \  `                                 |
    ||  \    -n         n                   |     ||  \      -n         n                   |     ||  \      -n         n                   |       ||  \    -n         n                   |
    ||  /   3  *(-2 + x)       otherwise    |     ||  /   n*3  *(-2 + x)       otherwise    |     ||  /   n*3  *(-2 + x)       otherwise    |       ||  /   3  *(-2 + x)       otherwise    |
    || /__,                                 |     || /__,                                   |     || /__,                                   |       || /__,                                 |
    \\n = 1                                 /     ||n = 1                                   |     ||n = 1                                   |       \\n = 1                                 /
                                                  \\                                        /     \\                                        /                                                
$$x \left(\begin{cases} \frac{\frac{x}{3} - \frac{2}{3}}{\left(\frac{5}{3} - \frac{x}{3}\right)^{2}} & \text{for}\: \left|{\frac{x}{3} - \frac{2}{3}}\right| < 1 \\\sum_{n=1}^{\infty} 3^{- n} n \left(x - 2\right)^{n} & \text{otherwise} \end{cases}\right) + 2 x \left(\begin{cases} \frac{\frac{x}{3} - \frac{2}{3}}{\frac{5}{3} - \frac{x}{3}} & \text{for}\: \left|{\frac{x}{3} - \frac{2}{3}}\right| < 1 \\\sum_{n=1}^{\infty} 3^{- n} \left(x - 2\right)^{n} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} \frac{\frac{x}{3} - \frac{2}{3}}{\left(\frac{5}{3} - \frac{x}{3}\right)^{2}} & \text{for}\: \left|{\frac{x}{3} - \frac{2}{3}}\right| < 1 \\\sum_{n=1}^{\infty} 3^{- n} n \left(x - 2\right)^{n} & \text{otherwise} \end{cases}\right) - 4 \left(\begin{cases} \frac{\frac{x}{3} - \frac{2}{3}}{\frac{5}{3} - \frac{x}{3}} & \text{for}\: \left|{\frac{x}{3} - \frac{2}{3}}\right| < 1 \\\sum_{n=1}^{\infty} 3^{- n} \left(x - 2\right)^{n} & \text{otherwise} \end{cases}\right)$$
-4*Piecewise(((-2/3 + x/3)/(5/3 - x/3), |-2/3 + x/3| < 1), (Sum(3^(-n)*(-2 + x)^n, (n, 1, oo)), True)) - 2*Piecewise(((-2/3 + x/3)/(5/3 - x/3)^2, |-2/3 + x/3| < 1), (Sum(n*3^(-n)*(-2 + x)^n, (n, 1, oo)), True)) + x*Piecewise(((-2/3 + x/3)/(5/3 - x/3)^2, |-2/3 + x/3| < 1), (Sum(n*3^(-n)*(-2 + x)^n, (n, 1, oo)), True)) + 2*x*Piecewise(((-2/3 + x/3)/(5/3 - x/3), |-2/3 + x/3| < 1), (Sum(3^(-n)*(-2 + x)^n, (n, 1, oo)), True))

    Ejemplos de hallazgo de la suma de la serie