combinatorics - Sum of infinite series $sum_{n=50}^{infty} frac{1}{binom{n}{50}}$

Find Sum of infinite series $$S=\sum_{n=50}^{\infty} \frac{1}{\binom{n}{50}}$$ My Try is :

$$50S=\sum_{n=50}^{\infty} \frac{n-(n-50)}{\binom{n}{50}}$$ so

$$50S=\sum_{n=50}^{\infty}\frac{n}{\binom{n}{50}}-\sum_{n=50}^{\infty}\frac{n-50}{\binom{n}{n-50}}$$ so

$$50S=\sum_{n=50}^{\infty}\frac{n}{\binom{n}{50}}-\sum_{n=0}^{\infty}\frac{n}{{\binom{n+50}{50}}}$$

any clue further

Answer

Hint:

$$\frac{1}{\binom{n}{50}} = \frac{50}{49} \bigg( \frac{1}{\binom{n-1}{49}} - \frac{1}{\binom{n}{49}} \bigg).$$