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



=

Solución

Ha introducido [src]
  oo            
____            
\   `           
 \         n    
  \     n*x    n
  /   -------*3 
 /    8*n - 9   
/___,           
n = 2           
$$\sum_{n=2}^{\infty} 3^{n} \frac{n x^{n}}{8 n - 9}$$
Sum(((n*x^n)/(8*n - 9))*3^n, (n, 2, oo))
Radio de convergencia de la serie de potencias
Se da una serie:
$$3^{n} \frac{n x^{n}}{8 n - 9}$$
Es la serie del tipo
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- serie de potencias.
El radio de convergencia de la serie de potencias puede calcularse por la fórmula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
En nuestro caso
$$a_{n} = \frac{3^{n} n}{8 n - 9}$$
y
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
entonces
$$R = \lim_{n \to \infty}\left(\frac{3^{n} 3^{- n - 1} n \left|{\frac{8 n - 1}{8 n - 9}}\right|}{n + 1}\right)$$
Tomamos como el límite
hallamos
$$R^{1} = \frac{1}{3}$$
$$R^{1} = 0.333333333333333$$
$$R = 0.333333333333333$$
Respuesta [src]
/      /                                                                                                   /                 pi*I\             -3*pi*I    /                 5*pi*I\             -pi*I     /                 7*pi*I\             pi*I    /                 pi*I\             3*pi*I    /                 3*pi*I\                 /                 3*pi*I\\               
|      |                                                                                                   |                 ----|             -------    |                 ------|             ------    |                 ------|             ----    |                 ----|             ------    |                 ------|                 |                 ------||               
|      |                  8 ___    /    8 ___ 8 ___\      8 ___    /    8 ___ 8 ___  pi*I\        8 ___    |    8 ___ 8 ___   2  |      8 ___     4       |    8 ___ 8 ___    4   |      8 ___    4       |    8 ___ 8 ___    4   |      8 ___   4      |    8 ___ 8 ___   4  |      8 ___    4       |    8 ___ 8 ___    4   |        8 ___    |    8 ___ 8 ___    2   ||               
|    2 |      7        21*\/ 3 *log\1 - \/ 3 *\/ x /   21*\/ 3 *log\1 - \/ 3 *\/ x *e    /   21*I*\/ 3 *log\1 - \/ 3 *\/ x *e    /   21*\/ 3 *e       *log\1 - \/ 3 *\/ x *e      /   21*\/ 3 *e      *log\1 - \/ 3 *\/ x *e      /   21*\/ 3 *e    *log\1 - \/ 3 *\/ x *e    /   21*\/ 3 *e      *log\1 - \/ 3 *\/ x *e      /   21*I*\/ 3 *log\1 - \/ 3 *\/ x *e      /|               
|18*x *|- ---------- - ----------------------------- + ----------------------------------- - ------------------------------------- - ---------------------------------------------- - --------------------------------------------- - ----------------------------------------- - --------------------------------------------- + ---------------------------------------|               
|      |  -16 + 48*x                   7/8                                7/8                                    7/8                                         7/8                                              7/8                                           7/8                                           7/8                                          7/8               |               
|      \                          128*x                              128*x                                  128*x                                       128*x                                            128*x                                         128*x                                         128*x                                        128*x                  /               
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------  for 3*|x| < 1
|                                                                                                                                                                                    7                                                                                                                                                                                                   
<                                                                                                                                                                                                                                                                                                                                                                                        
|                                                                                                                                                                               oo                                                                                                                                                                                                       
|                                                                                                                                                                             ____                                                                                                                                                                                                       
|                                                                                                                                                                             \   `                                                                                                                                                                                                      
|                                                                                                                                                                              \       n  n                                                                                                                                                                                              
|                                                                                                                                                                               \   n*3 *x                                                                                                                                                                                               
|                                                                                                                                                                               /   --------                                                                                                                                                                                  otherwise  
|                                                                                                                                                                              /    -9 + 8*n                                                                                                                                                                                             
|                                                                                                                                                                             /___,                                                                                                                                                                                                      
\                                                                                                                                                                             n = 2                                                                                                                                                                                                      
$$\begin{cases} \frac{18 x^{2} \left(- \frac{7}{48 x - 16} - \frac{21 \sqrt[8]{3} \log{\left(- \sqrt[8]{3} \sqrt[8]{x} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{21 \sqrt[8]{3} e^{\frac{i \pi}{4}} \log{\left(- \sqrt[8]{3} \sqrt[8]{x} e^{\frac{i \pi}{4}} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{21 \sqrt[8]{3} i \log{\left(- \sqrt[8]{3} \sqrt[8]{x} e^{\frac{i \pi}{2}} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{21 \sqrt[8]{3} e^{\frac{3 i \pi}{4}} \log{\left(- \sqrt[8]{3} \sqrt[8]{x} e^{\frac{3 i \pi}{4}} + 1 \right)}}{128 x^{\frac{7}{8}}} + \frac{21 \sqrt[8]{3} \log{\left(- \sqrt[8]{3} \sqrt[8]{x} e^{i \pi} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{21 \sqrt[8]{3} e^{- \frac{3 i \pi}{4}} \log{\left(- \sqrt[8]{3} \sqrt[8]{x} e^{\frac{5 i \pi}{4}} + 1 \right)}}{128 x^{\frac{7}{8}}} + \frac{21 \sqrt[8]{3} i \log{\left(- \sqrt[8]{3} \sqrt[8]{x} e^{\frac{3 i \pi}{2}} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{21 \sqrt[8]{3} e^{- \frac{i \pi}{4}} \log{\left(- \sqrt[8]{3} \sqrt[8]{x} e^{\frac{7 i \pi}{4}} + 1 \right)}}{128 x^{\frac{7}{8}}}\right)}{7} & \text{for}\: 3 \left|{x}\right| < 1 \\\sum_{n=2}^{\infty} \frac{3^{n} n x^{n}}{8 n - 9} & \text{otherwise} \end{cases}$$
Piecewise((18*x^2*(-7/(-16 + 48*x) - 21*3^(1/8)*log(1 - 3^(1/8)*x^(1/8))/(128*x^(7/8)) + 21*3^(1/8)*log(1 - 3^(1/8)*x^(1/8)*exp_polar(pi*i))/(128*x^(7/8)) - 21*i*3^(1/8)*log(1 - 3^(1/8)*x^(1/8)*exp_polar(pi*i/2))/(128*x^(7/8)) - 21*3^(1/8)*exp(-3*pi*i/4)*log(1 - 3^(1/8)*x^(1/8)*exp_polar(5*pi*i/4))/(128*x^(7/8)) - 21*3^(1/8)*exp(-pi*i/4)*log(1 - 3^(1/8)*x^(1/8)*exp_polar(7*pi*i/4))/(128*x^(7/8)) - 21*3^(1/8)*exp(pi*i/4)*log(1 - 3^(1/8)*x^(1/8)*exp_polar(pi*i/4))/(128*x^(7/8)) - 21*3^(1/8)*exp(3*pi*i/4)*log(1 - 3^(1/8)*x^(1/8)*exp_polar(3*pi*i/4))/(128*x^(7/8)) + 21*i*3^(1/8)*log(1 - 3^(1/8)*x^(1/8)*exp_polar(3*pi*i/2))/(128*x^(7/8)))/7, 3*|x| < 1), (Sum(n*3^n*x^n/(-9 + 8*n), (n, 2, oo)), True))

    Ejemplos de hallazgo de la suma de la serie