Wikipedias article about the binomial coefficient states:
"The series $${\displaystyle {\frac {k-1}{k}}\sum _{j=0}^{\infty }{\frac {1}{\binom {j+x}{k}}}={\frac {1}{\binom {x-1}{k-1}}}}$$ is convergent for $k ≥ 2$. This formula is used in the analysis of the German tank problem. It follows from $${\displaystyle {\frac {k-1}{k}}\sum _{j=0}^{M}{\frac {1}{\binom {j+x}{k}}}={\frac {1}{\binom {x-1}{k-1}}}-{\frac {1}{\binom {M+x}{k-1}}}}$$ which is proved by induction on $M$."
I am trying to find some meaning in the second formula. A proof through induction is fine, but it gives you little insight about where the formula comes from or what it means (combinatorically or as formula of probabilities).
Can anybody shed some light on this?