As it has a free parameter $λ$ Ι have tried to prove it by the WZ method. However it seems that the function is not hypergeometric in the symbols $n$ and $k$, that is $G(n+1,\lambda)/G(n,\lambda)$ and $G(n,\lambda+1)/G(n,\lambda)$ are not rational functions.
I have not seen this formula for $\pi$ before in the literature in spite that I know many other fórmulas. Although the series is not good for computing $\pi$ (contrary of the stated in the press) it could have some interest. In the paper there is also a fórmula of the same style for $\zeta(2)$.Taking the limit of it as $λ \to \infty$ we obtained a well known alternating series with a better convergence.