Polynomial long division: different answers when reordering terms

Polynomial long division: different answers when reordering terms

When I use polynomial long division to divide $\frac{1}{1-x}$, I get
$\;1 + x + x^2 +x^3 + x^4 + \cdots$
But when I just change the order of terms in the divisor:
$\frac{1}{-x+1}$, the long division algorithm gives me a very different
answer: $-\frac{1}{x} - \frac{1}{x^2} - \frac{1}{x^3} - \frac{1}{x^4} -
\cdots$, which seems somewhat strange to me, because sums are supposed
to be commutative, so it looks that these two answers should be
equivalent. Am I right?
But I cannot see how could these two answers be equivalent. Could it be
true that $-\frac{1}{x} - \frac{1}{x^2} - \frac{1}{x^3} - \frac{1}{x^4} -
\cdots \;=\; 1 + x + x^2 +x^3 + x^4 + \cdots$ ?
If that is the case, how can i "prove" it? Or how can I transform one form
into the other?
I already figured out that I can express $-\frac{1}{x} - \frac{1}{x^2} -
\frac{1}{x^3} - \frac{1}{x^4} - \cdots$ as negative exponents:
$-x^{-1} - x^{-2} - x^{-3} - x^{-4} - \cdots$ and then factor the $-1$'s
inside the exponents: $-x^{-1\cdot1} - x^{-1\cdot2} - x^{-1\cdot3} -
x^{-1\cdot4} - \cdots$ which can be seen as a power of a power:
$-(x^{-1})^1 - (x^{-1})^2 - (x^{-1})^3 - (x^{-1})^4 - \cdots$ and the
minus sign can be factored out too: $-\left[ (x^{-1})^1 + (x^{-1})^2 +
(x^{-1})^3 + (x^{-1})^4 + \cdots \right]$ which almost gives me the form
of the other series with raising positive exponents. But there's that
dangling minus sign in front of it, and the 0th power term is missing :-/
so I don't know how to massage it any further to get the other form of the
expansion. Any ideas?
Edit:
I know about Taylor series expansions and I can expand $\frac{1}{1-x}$
this way into $1 + x + x^2 + x^3 + \cdots$ (I don't know how to get
the other form this way, though).
Also, I know about the convergence condition for the infinite series,
$|x| < 1$ in the case of the first answer, and $|x| > 1$ or
$|\frac{1}{x}| < 1$ in the case of the second answer. I just didn't
expect that this could matter in my case, when I just reorder the terms in
the divisor, which shouldn't produce different answer.
I suspected that this could be somehow related to the fact that raising
powers of a fraction is the same as falling powers of its inverse. But I
cannot figure out why just one of these or the other is given, depending
on the order of terms in the long division, and this is what confuses me
the most.