A curious limit

Calculus Level 3

If \(a_n = \dbinom{2n}{n}^{-1}\) find \(\displaystyle\lim_{n \to \infty}{\left|\dfrac{a_{n+1}}{a_{n}}\right|}\).


Bonus: Is \(\displaystyle\sum_{n=1}^{\infty}{a_n}\) convergent? If yes, what is the sum?

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