sum of reciprocal numbers of combinations

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.