convergence divergence – I have to show that the double serie diverges

I have to show that the double serie $$sum_{k=1}^{infty} sum_{i=1}^{infty} ((frac{1}{k+1} cdot (frac{k}{k+1})^{i}) – (frac{1}{k+2} cdot (frac{k+1}{k+2})^{i}))$$ convergence to $-frac{1}{2}$.
Wolfram Alpha also says this series convergence to $-frac{1}{2}$. Unfortunately, I don’t know the calculation route.