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?

You're forgetting that psi^1 is the *correction* to psi^0, the unperturbed state. The full wave function to first order is psi^0 + psi^1.

thanks, i probably would waste hours until i would notice my mistake...

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