The are typically small compared to the oscillator frequencies . (Just to give you an idea, sec to sec for optical transitions in atoms, with similar proportionalities for the other relevant transitions.) That means that at most frequencies, is nearly real
Suppose we only have a few frequencies. Below the smallest , all the (real) terms in the sum are positive and Re . As we increase , one by one the terms in the sum become negative (in their real part) until beyond the highest frequency the entire sum and hence Re .
As we sweep past each ``pole'' (where the real part in the denominator of a single term is zero) that term increases rapidly in the real part, then dives through zero to become large and negative, then increases monotonically to zero. Meanwhile, its (usually small) imaginary part grows, reaching a peak just where the real part is zero (when is pure imaginary). In the vicinity of the pole, the contribution of this term can dominate the rest of the sum. We define: