Find the coefficient of x^r in the expansion, in ascending power of x of 1/(1-x)³ .
Thanks.
1/(1-x)³ = [1+(-x)]^(-3) The term which includes x^r is
(-3)(-4)(-5)...(-3-r+1)
----------------------------(-x)^r (product of r numbers on top)
r!
(-1)^r(3)(4)(5)(6)...(r+2)
=--------------------------------(-1)^r*(x^r)
(1)(2)(3)....r
(r+1)(r+2)
=------------(-1)^(2r)*(x^r) (-1 to the power of 2r is 1)
2
Therefore the coefficient of x^r is (r+1)(r+2)/2