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