Let $^nC_k:=\dfrac{n!}{k!(n-k)!}$ Please prove that,for all natural number $k≥2$, $\displaystyle\sum_{n＝k+1}^{\infty}\frac{1}{^nC_k}＝\frac{1}{k-1}$
I tried to prove by induction, but I cannot. I guess it is proved by using Tayler series for some function, but I cannot find the function.
That is known as the German tank problem, and is one of the fundamental Binomial Identities.