On some Series involving Reciprocals of $\left(\genfrac{}{}{0px}{}{2n}{n}\right)$ and the Catalan's Constant $G$

We investigate a class of combinatorial sums involving reciprocals of central binomial coefficients , employing generating functions as the primary solution technique to formulate and analyze series involving the Catalan’s constant. Using a direct approach, we derive new identities through integral techniques.