If $\alpha $ and $\beta $ are the roots of the equation ${ x }^{ 2 }+px+q=0$ and if the sum $\left( \alpha +\beta \right) x-\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } .{ x }^{ 2 }+\frac { { \alpha }^{ 3 }+{ \beta }^{ 3 } }{ 3 } .{ x }^{ 3 }-...$ exists then it is equal to

Using Vieta's formulas for x^2 + px + q = 0, the sum of roots is alpha + beta = -p and the product is alpha*beta = q. The terms alpha^n + beta^n can be written recursively, and the series is the expansion of log(1 + px + qx^2) using the logarithmic series log(1+y) = y - y^2/2 + y^3/3. However, substituting the roots into the polynomial means the argument becomes 1 - px + q. The sum evaluates to log(1 + px + qx^2).

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