Question

If $\alpha ,\beta$ are the roots of the equation $x^2-px+q=0,$ prove that
${\log }_e(1+px+q{x}^2)=(\alpha +\beta )x-\frac{{\alpha }^2+{\beta }^2}{2}{x}^2+\frac{{\alpha }^3+{\beta }^3}{3}{x}^3+...$

Answer 1

Right hand side$=[\alpha x-\frac{{\alpha }^2{x}^2}{2}+\frac{{\alpha }^3{x}^3}{3}-...]+[\beta x-\frac{{\beta }^2{x}^2}{2}+\frac{{\beta }^3{x}^3}{3}-...]$
$={\log }_e(1+\alpha x)+{\log }_e(1+\beta x)$
$={\log }_e(1+(\alpha +\beta )x+\alpha \beta x^2)$
$={\log }_e(1+px+q{x}^2)=$Left hand side
Here, we have used the faets $\alpha +\beta =p$ and $\alpha \beta =q.$ We know this from the given roots of the quadratic equation. We have also assumed that that both $|\alpha .x|<1$ and $\left|\beta x\right|<1.$