Summing Up Reciprocal of Central Binomial Coefficient

Calculus Level 5

$\large \sum_{n=1}^\infty \frac{1}{{2n \choose n}}$

Suppose the following infinite sum can be written as $\dfrac{a+\pi b\sqrt{c}}{d},$

where $$a,b,c$$ and $$d$$ are integers with $$b$$ and $$c$$ prime. Then compute the value of $$a+b+c+d$$,

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