Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$, where $\left(\binom{a}{b}\right)=\dfrac{(a!)!}{(b!)!\left((a-b)!\right)!}$
I tried using an easy method that worked for $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\binom{2014+k}{2014}}$, but it didn't work due to the nested gamma functions that appeared.