the official solution to this problem is (B). I completely understand why the sum includes only even contributions. my problem with this question is that the sum includes the member n=0, but according to the first order perturbation theory:
notice that the index i runs on all non-negative integers except for i=n. returning to the question at hand we see that the sum in (B) should really run on all n≠0, and not include the term n=0. but because the sum includes this term, it turns out that (B) is incorrect and thus choice (E) is the answer.
what am i missing?