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
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.