For $|x| < 1$,the coefficient of $x^3$ in the expansion of $\log(1+x+x^2)$ in ascending powers of $x$ is:

Let $\alpha$ and $\beta$ be the roots of $5x^2 - 3x - 1 = 0$. Then the expression $\left[ (\alpha + \beta)x - \left( \frac{\alpha^2 + \beta^2}{2} \right)x^2 + \left( \frac{\alpha^3 + \beta^3}{3} \right)x^3 - \dots \right]$ is equal to:

$\left( \frac{a - b}{a} \right) + \frac{1}{2} \left( \frac{a - b}{a} \right)^2 + \frac{1}{3} \left( \frac{a - b}{a} \right)^3 + \dots = $

If $y = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \infty$,then $x = $

$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{3 \cdot 4 \cdot 5} + \frac{1}{5 \cdot 6 \cdot 7} + \dots \infty = $

$\frac{1}{2} - \frac{1}{2 \cdot 2^2} + \frac{1}{3 \cdot 2^3} - \frac{1}{4 \cdot 2^4} + \ldots$ is equal to