The following is a proof of Newton-Girard formula using formal power series. Let $z$ be an indeterminate and $f(z)$ be the polynomial $$ 1-E_1z+\ldots+(-1)^nE_nz^n $$ Take log and differentiate both sides of the equation $$ f(z) = \prod_{i=1}^n (1-x_iz) $$

We obtain \begin{equation} f'(z)/f(z) = \sum_{i=1}^n \frac{-x_i}{1-x_iz}, \label{eq2} \end{equation}where $f'(z)$ is the derivative of $f(z)$ $$ f'(z) = -E_1 + 2E_2z - \ldots +(-1)^{n}nE_n z^{n-1}. $$

The right hand side of () is equal to $$ -\sum_{i=1}^n \sum_{k=0}^\infty x_i^{k+1} z^{k} = -\sum_{k=0}^\infty S_{k+1} z^{k}. $$

By equating coefficients of $$ f'(z) = -f(z)(S_1+S_2z+S_3z^2+\ldots) $$ we get the Newton-Girard formula.