I know how to use synthetic division to divide a polynomial by one of lower degree, but how does one divide a poly. by one of higher degree?

In particular, a text I am studying says

1 divided by x-1 =1 + x + x^2 + x^3 + ...

iff -1 < x < 1,

but I don't know how to find this out myself.

How do you divide 1 by x-1 ?