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  • Suma de la serie:
  • n*x^n
  • (n+2) (n+2)
  • (7/10)^n (7/10)^n
  • 1/(2n-1) 1/(2n-1)
  • Expresiones idénticas

  • x^n/(uno -x^2n)
  • x en el grado n dividir por (1 menos x al cuadrado n)
  • x en el grado n dividir por (uno menos x al cuadrado n)
  • xn/(1-x2n)
  • xn/1-x2n
  • x^n/(1-x²n)
  • x en el grado n/(1-x en el grado 2n)
  • x^n/1-x^2n
  • x^n dividir por (1-x^2n)
  • Expresiones semejantes

  • x^n/(1-x^(2*n))
  • x^n/(1+x^2n)

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



=

Solución

Ha introducido [src]
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____          
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  /        2  
 /    1 - x *n
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n = 1         
$$\sum_{n=1}^{\infty} \frac{x^{n}}{- n x^{2} + 1}$$
Sum(x^n/(1 - x^2*n), (n, 1, oo))
Radio de convergencia de la serie de potencias
Se da una serie:
$$\frac{x^{n}}{- n x^{2} + 1}$$
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
$$a_{n} = \frac{1}{- n x^{2} + 1}$$
y
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
entonces
$$R = \lim_{n \to \infty} \left|{\frac{x^{2} \left(n + 1\right) - 1}{n x^{2} - 1}}\right|$$
Tomamos como el límite
hallamos
$$R^{1} = 1$$
$$R = 1$$
Respuesta [src]
/         /            2\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
|         |      -1 + x |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
|-lerchphi|x, 1, -------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
|         |          2  |         /   /              /        2                    2         \                              /        2                    2         \                               2      2         \     /    /        2                    2         \                              /        2                    2         \                               2      2                   \     /    /        2                    2         \                              /        2                    2         \                               2      2                          /        2                    2         \                              /        2                    2         \                               2      2                 \\
|         \         x   /         |   |              |      im (x)               re (x)      | /         2          2   \   |      re (x)               im (x)      | /       2        2   \    4*im (x)*re (x)      |     |    |      im (x)               re (x)      | /         2          2   \   |      re (x)               im (x)      | /       2        2   \    4*im (x)*re (x)                |     |    |      im (x)               re (x)      | /         2          2   \   |      re (x)               im (x)      | /       2        2   \    4*im (x)*re (x)                       |      im (x)               re (x)      | /         2          2   \   |      re (x)               im (x)      | /       2        2   \    4*im (x)*re (x)              ||
|-------------------------  for Or|And||x| <= 1, 1 + |------------------ - ------------------|*\-1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\-1 + re (x) - im (x)/ - ------------------ < 0|, And|1 + |------------------ - ------------------|*\-1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\-1 + re (x) - im (x)/ - ------------------ >= 1, |x| < 1|, And|1 + |------------------ - ------------------|*\-1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\-1 + re (x) - im (x)/ - ------------------ >= 0, |x| <= 1, 1 + |------------------ - ------------------|*\-1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\-1 + re (x) - im (x)/ - ------------------ < 1, x != 1||
|            x                    |   |              |                 2                    2|                              |                 2                    2|                                           2    |     |    |                 2                    2|                              |                 2                    2|                                           2              |     |    |                 2                    2|                              |                 2                    2|                                           2                     |                 2                    2|                              |                 2                    2|                                           2            ||
|                                 |   |              |/  2        2   \    /  2        2   \ |                              |/  2        2   \    /  2        2   \ |                          /  2        2   \     |     |    |/  2        2   \    /  2        2   \ |                              |/  2        2   \    /  2        2   \ |                          /  2        2   \               |     |    |/  2        2   \    /  2        2   \ |                              |/  2        2   \    /  2        2   \ |                          /  2        2   \                      |/  2        2   \    /  2        2   \ |                              |/  2        2   \    /  2        2   \ |                          /  2        2   \             ||
|                                 \   \              \\im (x) + re (x)/    \im (x) + re (x)/ /                              \\im (x) + re (x)/    \im (x) + re (x)/ /                          \im (x) + re (x)/     /     \    \\im (x) + re (x)/    \im (x) + re (x)/ /                              \\im (x) + re (x)/    \im (x) + re (x)/ /                          \im (x) + re (x)/               /     \    \\im (x) + re (x)/    \im (x) + re (x)/ /                              \\im (x) + re (x)/    \im (x) + re (x)/ /                          \im (x) + re (x)/                      \\im (x) + re (x)/    \im (x) + re (x)/ /                              \\im (x) + re (x)/    \im (x) + re (x)/ /                          \im (x) + re (x)/             //
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
<       oo                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
|     ____                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
|     \   `                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
|      \        n                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               
|       \      x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
|        )  --------                                                                                                                                                                                                                                                                                                                                                                             otherwise                                                                                                                                                                                                                                                                                                                                                                      
|       /          2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|      /    1 - n*x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
|     /___,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
|     n = 1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
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$$\begin{cases} - \frac{\Phi\left(x, 1, \frac{x^{2} - 1}{x^{2}}\right)}{x} & \text{for}\: \left(\left|{x}\right| \leq 1 \wedge \left(- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(2 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(x\right)}\right)^{2} - 1\right) + \left(\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 1\right) + 1 - \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} < 0\right) \vee \left(\left(- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(2 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(x\right)}\right)^{2} - 1\right) + \left(\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 1\right) + 1 - \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} \geq 1 \wedge \left|{x}\right| < 1\right) \vee \left(\left(- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(2 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(x\right)}\right)^{2} - 1\right) + \left(\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 1\right) + 1 - \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} \geq 0 \wedge \left|{x}\right| \leq 1 \wedge \left(- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(2 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(x\right)}\right)^{2} - 1\right) + \left(\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 1\right) + 1 - \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} < 1 \wedge x \neq 1\right) \\\sum_{n=1}^{\infty} \frac{x^{n}}{- n x^{2} + 1} & \text{otherwise} \end{cases}$$
Piecewise((-lerchphi(x, 1, (-1 + x^2)/x^2)/x, ((|x| <= 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(-1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(-1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 < 0))∨((|x| < 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(-1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(-1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 >= 1))∨((Ne(x, 1))∧(|x| <= 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(-1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(-1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 >= 0)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(-1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(-1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 < 1))), (Sum(x^n/(1 - n*x^2), (n, 1, oo)), True))

    Ejemplos de hallazgo de la suma de la serie