It is a frequently unacknowledged yet extremely useful fact that the binomial theorem works for all powers of binomials, not just integer powers. That is, for any a,
(1+x)a = 1 + ax + a(a-1)/2!x2 + ... + a(a-1)...(a-k+1)/k!xk + ...
Knowing this, find with proof the coefficient of xn in the binomial expansion of (1+x)½ for n>=2.
Provide an answer in closed form. That is, n! is acceptable, while n(n-1)(n-2)...(3)(2)(1) is not. |
