I am working through the Griffiths QM text and I am getting caught up on some the process he uses to derive the wave functions and energy levels for the QHO, via Frobenius/Power series method. I understand that the Schrodinger equation get recast in to a summation form over the coefficients of the power series expansion. I understand that each of the terms in the summation, which represents the original Schrodinger equation for the QHO, must be equal to zero. This is the replacement for the differential equation. Once this is realized the recursion formula for all of the coefficients of the wave function is produced. Griffiths breaks this down into odd and even functions for the values of the index j. Discusses the divergence of the series, which was similar to asymptotic expansion he introduced at the beginning of the section. He then explains that the way to make the series convergent is by requiring the series to terminate at j=n. This imposes the condition on E that solves for the energy levels of the QHO. What I am confused by is why, when I set n=0, I must set the first odd coefficient to zero. Does it have to due with the physicality of the wave function? On the same note, I am confused with larger values of n and this same sort of thing. Lets say that I choose the n=2 state, so that the series will be cutoff at n=2, if I choose the first odd-coefficient to be zero wouldn't I loose some portion of the wave function that could be important? How do we know that if n is even the odd terms must disappear and vice-versa? I realize that the wave functions for the QHO should be alternatively odd and even functions. i am missing something. Thanks ahead of time for whatever help is offered.