\begin{array}{rcl} \input amstex
\displaystyle\sum_{k=0}^\infty\frac{1}{\dbinom{n+k}{k}}&=&\displaystyle\sum_{k=0}^\infty\frac{n!\;k!}{(n+k)!}\\
&=&\displaystyle\sum_{k=0}^\infty\frac{n\;\Gamma(n)\;\Gamma(k+1) }{\Gamma(n+k+1)}\\
&=&n\displaystyle\sum_{k=0}^\infty\textbf B(k+1,n) \\
&=&n\displaystyle\sum_{k=0}^\infty\int_{0}^{1}x^{k}\;(1-x)^{n-1}\;\textbf{d}x\\
&=&n\displaystyle\int\limits_{0}^{1}(1-x)^{n-1}\left(\sum_{k=0}^\infty\;x^{k}\right)\;\textbf{d}x \\
&=&n\displaystyle\int\limits_{0}^{1}(1-x)^{n-2}\;\textbf{d}x \\
&=&-n\displaystyle\int\limits_{0}^{1}(1-x)^{n-2}\;\textbf{d}(1-x)\\
&=&-n\;\displaystyle\frac{(1-x)^{n-1}}{n-1}\bigg|_{0}^{1}\\
&=&\displaystyle\frac{n}{n-1}