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

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


∑
=

Solución

Ha introducido [src] 
  oo          
____          
\   `         
 \        n   
  \      x    
   )  --------
  /        2  
 /    1 + x *n
/___,         
n = 1
∑n=1∞xnnx2+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:
xnnx2+1\frac{x^{n}}{n x^{2} + 1}
Es la serie del tipo
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
En nuestro caso
an=1nx2+1a_{n} = \frac{1}{n x^{2} + 1}
y
x0=0x_{0} = 0
,
d=1d = 1
,
c=1c = 1
entonces
R=lim⁡n→∞∣x2(n+1)+1nx2+1∣R = \lim_{n \to \infty} \left|{\frac{x^{2} \left(n + 1\right) + 1}{n x^{2} + 1}}\right|
Tomamos como el límite
hallamos
R1=1R^{1} = 1
R=1R = 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)/             //
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<      oo                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|    ____                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
|    \   `                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
|     \        n                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
|      \      x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
|       )  --------                                                                                                                                                                                                                                                                                                                                                                       otherwise                                                                                                                                                                                                                                                                                                                                                                  
|      /          2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
|     /    1 + n*x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
|    /___,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
|    n = 1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
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{Φ(x,1,x2+1x2)xfor (∣x∣≤1∧(−(re⁡(x))2((re⁡(x))2+(im⁡(x))2)2+(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2)(2(re⁡(x))2−2(im⁡(x))2+1)+((re⁡(x))2((re⁡(x))2+(im⁡(x))2)2−(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2)((re⁡(x))2−(im⁡(x))2+1)+1−4(re⁡(x))2(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2<0)∨((−(re⁡(x))2((re⁡(x))2+(im⁡(x))2)2+(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2)(2(re⁡(x))2−2(im⁡(x))2+1)+((re⁡(x))2((re⁡(x))2+(im⁡(x))2)2−(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2)((re⁡(x))2−(im⁡(x))2+1)+1−4(re⁡(x))2(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2≥1∧∣x∣<1)∨((−(re⁡(x))2((re⁡(x))2+(im⁡(x))2)2+(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2)(2(re⁡(x))2−2(im⁡(x))2+1)+((re⁡(x))2((re⁡(x))2+(im⁡(x))2)2−(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2)((re⁡(x))2−(im⁡(x))2+1)+1−4(re⁡(x))2(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2≥0∧∣x∣≤1∧(−(re⁡(x))2((re⁡(x))2+(im⁡(x))2)2+(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2)(2(re⁡(x))2−2(im⁡(x))2+1)+((re⁡(x))2((re⁡(x))2+(im⁡(x))2)2−(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2)((re⁡(x))2−(im⁡(x))2+1)+1−4(re⁡(x))2(im⁡(x))2((re⁡(x))2+(im⁡(x))2)2<1∧x≠1)∑n=1∞xnnx2+1otherwise\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

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