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  • Suma de la serie:
  • 1/n 1/n
  • 1/n(n+1) 1/n(n+1)
  • (-1)^n/n^2 (-1)^n/n^2
  • 1/(2^n) 1/(2^n)
  • Expresiones idénticas

  • nx^n/((8n- nueve)* tres ^n)
  • nx en el grado n dividir por ((8n menos 9) multiplicar por 3 en el grado n)
  • nx en el grado n dividir por ((8n menos nueve) multiplicar por tres en el grado n)
  • nxn/((8n-9)*3n)
  • nxn/8n-9*3n
  • nx^n/((8n-9)3^n)
  • nxn/((8n-9)3n)
  • nxn/8n-93n
  • nx^n/8n-93^n
  • nx^n dividir por ((8n-9)*3^n)
  • Expresiones semejantes

  • nx^n/((8n+9)*3^n)
  • Expresiones con funciones

  • nx
  • nx^n/3^n(n+1)
  • nx^n/(n+2)
  • nx^n/4^n
  • nx^6n
  • nxn^2

Suma de la serie nx^n/((8n-9)*3^n)



=

Solución

Ha introducido [src]
  oo              
____              
\   `             
 \           n    
  \       n*x     
   )  ------------
  /              n
 /    (8*n - 9)*3 
/___,             
n = 2             
$$\sum_{n=2}^{\infty} \frac{n x^{n}}{3^{n} \left(8 n - 9\right)}$$
Sum((n*x^n)/(((8*n - 9)*3^n)), (n, 2, oo))
Respuesta [src]
/     /                                                                                              /                pi*I\                       /                5*pi*I\                      /                7*pi*I\                    /                pi*I\                      /                3*pi*I\                /                3*pi*I\\             
|     |                                                                                              |                ----|            -3*pi*I    |                ------|            -pi*I     |                ------|            pi*I    |                ----|            3*pi*I    |                ------|                |                ------||             
|     |                          /     7/8 8 ___\              /     7/8 8 ___  pi*I\                |     7/8 8 ___   2  |            -------    |     7/8 8 ___    4   |            ------    |     7/8 8 ___    4   |            ----    |     7/8 8 ___   4  |            ------    |     7/8 8 ___    4   |                |     7/8 8 ___    2   ||             
|     |                   7/8    |    3   *\/ x |       7/8    |    3   *\/ x *e    |         7/8    |    3   *\/ x *e    |       7/8     4       |    3   *\/ x *e      |       7/8    4       |    3   *\/ x *e      |       7/8   4      |    3   *\/ x *e    |       7/8    4       |    3   *\/ x *e      |         7/8    |    3   *\/ x *e      ||             
|     |               63*3   *log|1 - ----------|   63*3   *log|1 - ----------------|   63*I*3   *log|1 - ----------------|   63*3   *e       *log|1 - ------------------|   63*3   *e      *log|1 - ------------------|   63*3   *e    *log|1 - ----------------|   63*3   *e      *log|1 - ------------------|   63*I*3   *log|1 - ------------------||             
|   2 |      7                   \        3     /              \           3        /                \           3        /                       \            3         /                      \            3         /                    \           3        /                      \            3         /                \            3         /|             
|2*x *|- ---------- - --------------------------- + --------------------------------- - ----------------------------------- - -------------------------------------------- - ------------------------------------------- - --------------------------------------- - ------------------------------------------- + -------------------------------------|             
|     |        16*x                  7/8                              7/8                                  7/8                                       7/8                                            7/8                                         7/8                                         7/8                                        7/8              |             
|     |  -16 + ----             128*x                            128*x                                128*x                                     128*x                                          128*x                                       128*x                                       128*x                                      128*x                 |             
|     \         3                                                                                                                                                                                                                                                                                                                                       /      |x|    
|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------  for --- < 1
<                                                                                                                                                                           63                                                                                                                                                                                  3     
|                                                                                                                                                                                                                                                                                                                                                                     
|                                                                                                                                                                       oo                                                                                                                                                                                            
|                                                                                                                                                                     ____                                                                                                                                                                                            
|                                                                                                                                                                     \   `                                                                                                                                                                                           
|                                                                                                                                                                      \       -n  n                                                                                                                                                                                  
|                                                                                                                                                                       \   n*3  *x                                                                                                                                                                                   
|                                                                                                                                                                       /   --------                                                                                                                                                                        otherwise 
|                                                                                                                                                                      /    -9 + 8*n                                                                                                                                                                                  
|                                                                                                                                                                     /___,                                                                                                                                                                                           
|                                                                                                                                                                     n = 2                                                                                                                                                                                           
\                                                                                                                                                                                                                                                                                                                                                                     
$$\begin{cases} \frac{2 x^{2} \left(- \frac{7}{\frac{16 x}{3} - 16} - \frac{63 \cdot 3^{\frac{7}{8}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} e^{\frac{i \pi}{4}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{i \pi}{4}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} i \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{i \pi}{2}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} e^{\frac{3 i \pi}{4}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{3 i \pi}{4}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} + \frac{63 \cdot 3^{\frac{7}{8}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{i \pi}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} e^{- \frac{3 i \pi}{4}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{5 i \pi}{4}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} + \frac{63 \cdot 3^{\frac{7}{8}} i \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{3 i \pi}{2}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} e^{- \frac{i \pi}{4}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{7 i \pi}{4}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}}\right)}{63} & \text{for}\: \frac{\left|{x}\right|}{3} < 1 \\\sum_{n=2}^{\infty} \frac{3^{- n} n x^{n}}{8 n - 9} & \text{otherwise} \end{cases}$$
Piecewise((2*x^2*(-7/(-16 + 16*x/3) - 63*3^(7/8)*log(1 - 3^(7/8)*x^(1/8)/3)/(128*x^(7/8)) + 63*3^(7/8)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(pi*i)/3)/(128*x^(7/8)) - 63*i*3^(7/8)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(pi*i/2)/3)/(128*x^(7/8)) - 63*3^(7/8)*exp(-3*pi*i/4)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(5*pi*i/4)/3)/(128*x^(7/8)) - 63*3^(7/8)*exp(-pi*i/4)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(7*pi*i/4)/3)/(128*x^(7/8)) - 63*3^(7/8)*exp(pi*i/4)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(pi*i/4)/3)/(128*x^(7/8)) - 63*3^(7/8)*exp(3*pi*i/4)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(3*pi*i/4)/3)/(128*x^(7/8)) + 63*i*3^(7/8)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(3*pi*i/2)/3)/(128*x^(7/8)))/63, |x|/3 < 1), (Sum(n*3^(-n)*x^n/(-9 + 8*n), (n, 2, oo)), True))

    Ejemplos de hallazgo de la suma de la serie