Catalan's constant with consecutive numbers

A little tweak of a special type on three consecutive numbers and it is great as Dirichlet beta function.

And I have used it below to get an expression of the Catalan constant.

So all in all with the consecutive numbers model, I have managed to get expressions for 1, root 2, pi, e, golden ratio, Euler Mascheroni constant, Apery Constant and Catalan constant.

Have a nice day !!