Each fraction 1/(x-a) can be written as a/(x/a-1).

Use the power series representation of 1/(1+u) as 1+u+u²+u³+... to represent 1/(x-a) as a power series around x=0.

Aside: recall that the power (i.e, Taylor) series representation around zero is called the MacLaurin series.
1/(x-a) has a different power series around x=infinity.