What is the value of the sum?

Consider the following infinite series:

$\frac{1}{1+z} = 1 - z + z^2 -z^3 + \dots$

Where $\mid z\mid < 1$

Integrating both sides from $0$ to $z$ gives:

$\ln(1+z) = z - \frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} + \dots$

Where $\mid z\mid < 1$

Now the given series can be rewritten as:

$S = \left((\alpha x) - \frac{(\alpha x)^2}{2} + \frac{(\alpha x)^3}{3} - \dots\right) + \left((\beta x) - \frac{(\beta x)^2}{2} + \frac{(\beta x)^3}{3} - \dots\right)$

Now, the pieces of information missing in the problem statement are that $\mid \alpha x\mid < 1$ and $\mid \beta x \mid < 1$ . Assuming that these two conditions hold, the above-derived formulas are applied, leading to:

$S = \ln(1+\alpha x) + \ln(1+ \beta x)$

$S = \ln(1+ (\alpha + \beta)x + \alpha \beta x^2)$

Using the given information in the problem statement, it can be deduced that the sum and product of roots of the given quadratic equation are:

$\alpha + \beta = -p$ $\alpha \beta = q$

Therefore, the result is finally,

$\boxed{S = \ln(1 -px + qx^2)}$

The information which needs to be added to the problem statement are the conditions that $\mid \alpha x\mid < 1$ and $\mid \beta x \mid < 1$ .