Complex power term series

Determine whether the series ${\left(\frac{2^2}{1^2}-\frac{2}{1}\right)}^{-1}+{\left(\frac{3^3}{2^3}-\frac{3}{2}\right)}^{-2}+{\left(\frac{4^4}{3^4}-\frac{4}{3}\right)}^{-3}+\dots$ is convergent or divergent.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXI. Convergency and Divergency of Series, Examples XXI.a, No. 15