Negative Binomial Coefficient

While reading through a book on probability and calculating the expectation of geometric series, I came across an interesting sum
We know that if then the sum converges to


Coming from the other side and using binomial theorem we getwhich expands to



Which leads to an interesting conclusion

I got maxima to verify this by calling the binomial function for various combinations of -1 and n. I am a little confused about the interpretation of the meaning though

Usually nC2 is used to imply from "n" objects, choose 2 at a time. nC2 gives us the number of total such combinations. What does -1Cn mean? What are the other proofs for -1Cn?

binomial coefficient