Catalan's constant is irrational

Whether the constant $$G=\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{(2k+1{)}^2}=\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots$$ introduced by Catalan in the nineteen century is irrational, is a long-standing open problem. In this paper we prove the irrationality of $G$ via using suitable weights.