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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.

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That is known as the German tank problem, and is one of the fundamental Binomial Identities.

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