One of those...

\[\large A = \frac{\binom{2n + 1}{n + 1}}{\binom{2n - 1}{n}}\] has a closed form if \(n \in \mathbb{Z}^+\). Find this closed form, and give the value of \(\large A\) if \(n = 10^9\). Give your answer to 6 decimal places

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