In considering the physical admissibility of the solutions
we first noted that if 
is a finite
polynomial that 
is normalizable because of the extremely
rapid fall of the Gaussian factor. We then assumed that if the
series for 
did not terminate, that the resulting 
would
grow so rapidly that even the Gaussian decay of the factor
could not make 
normalizable. In this
subsection, we shall investigate why this should be the case.
The key observation which we shall exploit is that for large n the
coefficients 
in the sequence defining 
behave
much like
the terms in the expansion for 
so that, for large
X, 
grows at least like 
, so that not even
multiplication by
the Gaussian decay factor 
will render the wave function
normalizable. The basic similarity of the two expansions comes from
the behavior of the ratio of the 
to the 
terms in
the sequence successive terms.
For 
the ratio is
For 
the ratio approaches the same limit,
We find precisely the same limit in the case of our auxiliary
function, 
. For
the ratio is
Note that the ratio of successive terms may be negative for small
values of n so that the terms in the series for the 
may
alternate in sign. Because the ratio approaches a positive limit,
However, we are assured that for sufficiently large n, all terms
will be of the same sign. Moreover, because
, we may find an N so that
for all n>N the ratio of successive terms in 
will
exceed 
where 
is any number between
zero and one. This is important because this is just the quantity
(
, which is the ratio of successive terms in 
. If we now multiply 
by a constant 
so that the
term of 
is equal in magnitude to the
term of 
, this implies that all successive terms
in 
will exceed their counterparts in 
.
Because all of the terms in question are of the same sign, the
absolute value of the sum of all remaining terms in 
exceeds that of the associated terms in 
, which is
important because we know that g grows too fast to yield a
normalizable function. To complete the argument, let 
be the
sum of the first N terms in 
and 
be the sum of the
first N terms in 
. Note that both 
and 
are finite polynomials. We have just shown that for every 
there exist a constant 
and integer N such that for all
X,
where 
is a function which grows at most as quickly as an
order polynomial. If we now consider the behavior of the
wave function at large values of X, 
. (The limit of any
polynomial times a Gaussian factor is zero.) This statement is true
of any 
. For this to be true for 
, we
see that, as long as the series for 
does not terminate, we must
have 
as 
.
This completes the proof of our claim that we must reject all
solutions of 
for which the series do not terminate and
culminates our power series analysis of the quantum states of the SHO.