TITLE: Why can't partial fractions expansion be "normally" done is this case?
QUESTION [5 upvotes]: I've learned partial fractions but I couldn't really understand one thing. When we have a case when one of the factors has multiplicity $> 1$, we got to make a kind of "stairs". e.g.
$$\frac{1}{(x-1)(x+2)^2} = \frac{A}{(x-1)} + \frac{B}{(x+2)} + \frac{C}{(x+2)^2}.$$
But I thought $why$ couldn't it be written like this:
$$\frac{1}{(x-1)(x+2)^2} =\frac{A}{(x-1)} + \frac{B}{(x+2)^2}.$$
After some pencil and paper working, I realized it doesn't seem to work (and I don't know why!) I've been trying this for weeks, but I've got nowhere. So, if you could help me visualize a reason, I'd be grateful.
Thank you.

REPLY [2 votes]: Hint: every proper fraction of form $f(x)/x^n$ can be written as $(c_0 + c_1 x +\:\cdots\: c_{n-1} x^{n-1})/x^{n}$. Such proper fractions form an $n$ dimensional vector space over $\mathbb R$ with basis $x^{-1},\:x^{-2},\:\cdots\:x^{-n}\:$. Therefore if $n > 1$ then the single fraction $1/x^n$ does not span the whole space. The same holds true if one applies a shift automorphism $x\mapsto x+a$.