You're close, except you forgot to add a coefficient, rational functions have this relation:

It's that simple, now you have to find the coefficients:

Multiply the denominator through:

Now we multiply the sum through:

Adjust the indices:

Now we combine like terms:

Since there are no coefficients on the left side, then:

Since the coefficient of the x term on the left side is 1, then the coefficient of the x term on the right side is also 1:

This gives you:

Which gives you an interesting result:

Now you have your power series:

This gives you: