Series of Binomial Coefficients

I'm trying to solve an infinite series (it's part of a larger applied problem I'm working with, but I know what to do with this once I find it, so it's not relevant).

I need the sum from n = k to n = infinity of

C(n,k)*c^(-n)

for a positive integer c > 1.

It's a little different, sort of a reverse of the binomial theorem since the n runs with the series with k held constant.

I have found that

C(n+j,k) = C(n,k)*(n+j)!(n-k)!/[n!(n+j-k)!]

But that seemed to cause more harm than good...

I know from Mathematica the sum is going to be c/(c-1)^(k+1), but I'm having lots of trouble proving this.

Thanks in advance for any assistance.