[f97tosc]

sum{n = k to infinity} C(n,k)*x^n =
= {reindex} =

sum{n = 0 to infinity} C(n+k,k)*x^(n+k) =

x^k sum{n = 0 to infinity} C(n+k,k)*x^n =

x^k sum{n = 0 to infinity} C(n+k,(n+k)-k)*x^n =

x^k [sum{n = 0 to infinity} C(n+k,n)*x^n ]

I hadn't seen the sum in the brackets before but it was listed at Wikipedia to be:

sum{n = 0 to infinity} C(n+k,n)*x^n = 1/(1-x)^(k+1)

The stated result follows.