Let me start from a very simple illustrative example.
Say, we want to decompose
The correct procedure calls for a representation
which results in
and the answer
that cannot be solved for
The idea behind having members of the decomposition with every power of a denominator from 1 to maximum is to be able to accommodate any polynomial in the numerator that can contain a variable
that results in
which can be solved as
Without all three components we would not be able to accommodate any polynomial in the numerator. But, if the numerator does not contain a certain power of
But it's a very peculiar case and, when talking about general rule, it's advisable to retain members with all possible powers of the members of the denominator, from 1 to maximum.