Consider the function \( f(x) = \frac{1}{1 - x} \). Find the Taylor series expansion of \( f(x) \) up to and including the term \( x^3 \). Then, for the polynomial \( P(x) = x^3 + ax^2 + bx + c \), use Vieta's formulas to determine the coefficients \( a, b, \) and \( c \) if the roots of \( P(x) \) are the first three positive integers. Finally, compute the value of \( \sum_{i=1}^3 f(\alpha_i) \), where \( \alpha_i \) are the roots of \( P(x) \). Let's think step by step and output the final answer within \boxed{}.
user