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

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


∑
=

Solución

Ha introducido [src] 
  oo                        
____                        
\   `                       
 \         n         2*n + 1
  \    (-1)   /x - 2\       
  /   -------*|-----|       
 /    2*n - 1 \  3  /       
/___,                       
n = 1
∑n=1∞(−1)n2n−1(x−23)2n+1\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n}}{2 n - 1} \left(\frac{x - 2}{3}\right)^{2 n + 1} 
Sum(((-1)^n/(2*n - 1))*((x - 2)/3)^(2*n + 1), (n, 1, oo))
Radio de convergencia de la serie de potencias
Se da una serie:
(−1)n2n−1(x−23)2n+1\frac{\left(-1\right)^{n}}{2 n - 1} \left(\frac{x - 2}{3}\right)^{2 n + 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=(x3−23)2n+12n−1a_{n} = \frac{\left(\frac{x}{3} - \frac{2}{3}\right)^{2 n + 1}}{2 n - 1}
y
x0=1x_{0} = 1
,
d=1d = 1
,
c=0c = 0
entonces
R=∞~(1+lim⁡n→∞((2n+1)∣(x3−23)−2n−3(x3−23)2n+12n−1∣))R = \tilde{\infty} \left(1 + \lim_{n \to \infty}\left(\left(2 n + 1\right) \left|{\frac{\left(\frac{x}{3} - \frac{2}{3}\right)^{- 2 n - 3} \left(\frac{x}{3} - \frac{2}{3}\right)^{2 n + 1}}{2 n - 1}}\right|\right)\right)
Tomamos como el límite
hallamos
R1=∞~(1+9x2sign⁡(x2−4x+4)−4xsign⁡(x2−4x+4)+4sign⁡(x2−4x+4))R^{1} = \tilde{\infty} \left(1 + \frac{9}{x^{2} \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)} - 4 x \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)} + 4 \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)}}\right)
R1=∞~(1+9x2sign⁡(x2−4x+4)−4xsign⁡(x2−4x+4)+4sign⁡(x2−4x+4))R^{1} = \tilde{\infty} \left(1 + \frac{9}{x^{2} \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)} - 4 x \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)} + 4 \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)}}\right)
R=∞~(1+9x2sign⁡(x2−4x+4)−4xsign⁡(x2−4x+4)+4sign⁡(x2−4x+4))R = \tilde{\infty} \left(1 + \frac{9}{x^{2} \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)} - 4 x \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)} + 4 \operatorname{sign}{\left(x^{2} - 4 x + 4 \right)}}\right)
Respuesta [src] 
    //               /   ___________\                           \     //               /   ___________\                           \
    ||               |  /         2 |                           |     ||               |  /         2 |                           |
    ||         2     |\/  (-2 + x)  |                           |     ||         2     |\/  (-2 + x)  |                           |
    ||-(-2 + x) *atan|--------------|                           |     ||-(-2 + x) *atan|--------------|                           |
    ||               \      3       /                           |     ||               \      3       /                           |
    ||--------------------------------  for And(x >= -1, x <= 5)|     ||--------------------------------  for And(x >= -1, x <= 5)|
    ||             ___________                                  |     ||             ___________                                  |
    ||            /         2                                   |     ||            /         2                                   |
    ||        3*\/  (-2 + x)                                    |     ||        3*\/  (-2 + x)                                    |
  2*|<                                                          |   x*|<                                                          |
    ||   oo                                                     |     ||   oo                                                     |
    || ____                                                     |     || ____                                                     |
    || \   `                                                    |     || \   `                                                    |
    ||  \        n  -2*n         2*n                            |     ||  \        n  -2*n         2*n                            |
    ||   \   (-1) *3    *(-2 + x)                               |     ||   \   (-1) *3    *(-2 + x)                               |
    ||   /   -----------------------           otherwise        |     ||   /   -----------------------           otherwise        |
    ||  /            -1 + 2*n                                   |     ||  /            -1 + 2*n                                   |
    || /___,                                                    |     || /___,                                                    |
    \\ n = 1                                                    /     \\ n = 1                                                    /
- --------------------------------------------------------------- + ---------------------------------------------------------------
                                 3                                                                 3
x({−(x−2)2atan⁡((x−2)23)3(x−2)2for x≥−1∧x≤5∑n=1∞(−1)n3−2n(x−2)2n2n−1otherwise)3−2({−(x−2)2atan⁡((x−2)23)3(x−2)2for x≥−1∧x≤5∑n=1∞(−1)n3−2n(x−2)2n2n−1otherwise)3\frac{x \left(\begin{cases} - \frac{\left(x - 2\right)^{2} \operatorname{atan}{\left(\frac{\sqrt{\left(x - 2\right)^{2}}}{3} \right)}}{3 \sqrt{\left(x - 2\right)^{2}}} & \text{for}\: x \geq -1 \wedge x \leq 5 \\\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n} 3^{- 2 n} \left(x - 2\right)^{2 n}}{2 n - 1} & \text{otherwise} \end{cases}\right)}{3} - \frac{2 \left(\begin{cases} - \frac{\left(x - 2\right)^{2} \operatorname{atan}{\left(\frac{\sqrt{\left(x - 2\right)^{2}}}{3} \right)}}{3 \sqrt{\left(x - 2\right)^{2}}} & \text{for}\: x \geq -1 \wedge x \leq 5 \\\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n} 3^{- 2 n} \left(x - 2\right)^{2 n}}{2 n - 1} & \text{otherwise} \end{cases}\right)}{3} 
-2*Piecewise((-(-2 + x)^2*atan(sqrt((-2 + x)^2)/3)/(3*sqrt((-2 + x)^2)), (x >= -1)∧(x <= 5)), (Sum((-1)^n*3^(-2*n)*(-2 + x)^(2*n)/(-1 + 2*n), (n, 1, oo)), True))/3 + x*Piecewise((-(-2 + x)^2*atan(sqrt((-2 + x)^2)/3)/(3*sqrt((-2 + x)^2)), (x >= -1)∧(x <= 5)), (Sum((-1)^n*3^(-2*n)*(-2 + x)^(2*n)/(-1 + 2*n), (n, 1, oo)), True))/3

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