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



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Solución

Ha introducido [src]
  oo                          
____                          
\   `                         
 \        n + 1  n + 1        
  \   (-1)     *x             
  /   ----------------*(n + 1)
 /           n                
/___,                         
n = 1                         
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1} x^{n + 1}}{n} \left(n + 1\right)$$
Sum((((-1)^(n + 1)*x^(n + 1))/n)*(n + 1), (n, 1, oo))
Respuesta [src]
    // -log(1 + x)    for And(x <= 1, x > -1)\     //     -x                    \
    ||                                       |     ||    -----       for |x| < 1|
    ||  oo                                   |     ||    1 + x                  |
    ||____                                   |     ||                           |
    ||\   `                                  |     ||  oo                       |
- x*|< \        n  n                         | - x*|< ___                       |
    ||  \   (-1) *x                          |     || \  `                      |
    ||  /   --------         otherwise       |     ||  \       n  n             |
    || /       n                             |     ||  /   (-1) *x    otherwise |
    ||/___,                                  |     || /__,                      |
    \\n = 1                                  /     \\n = 1                      /
$$- x \left(\begin{cases} - \frac{x}{x + 1} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=1}^{\infty} \left(-1\right)^{n} x^{n} & \text{otherwise} \end{cases}\right) - x \left(\begin{cases} - \log{\left(x + 1 \right)} & \text{for}\: x \leq 1 \wedge x > -1 \\\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n} x^{n}}{n} & \text{otherwise} \end{cases}\right)$$
-x*Piecewise((-log(1 + x), (x <= 1)∧(x > -1)), (Sum((-1)^n*x^n/n, (n, 1, oo)), True)) - x*Piecewise((-x/(1 + x), |x| < 1), (Sum((-1)^n*x^n, (n, 1, oo)), True))

    Ejemplos de hallazgo de la suma de la serie