Multiplying series is extra work in this case.
Notice that and ...
1. Multiply the series for (1-x)^-1 (valid when |x|<1) by itself and so obtain the expansion
(1-x)^-2 = sigma n=0 to infinity (n+1)x^n, |x|<1.
2. Derive the expansion
(1-x)^-3 = sigma n=0 to infinity ((n+1)(n+2)*x^n)/2 (|x|<1)
by multiplication of series. Use the result of Exercise 1.