A typical problem in physics is that physical quantities can be calculated for a series of finite systems, but ultimately, we would like to know the results for the infinite system. Hulthen (1938) studied the one-dimensional Spin Heisenberg model with the Hamiltonian

Question

where is the spin at the site and is the total number of sites in the system. From the eigen equation

Hulthen obtained the ground-state energy per site, for a series of finite systems with

Now assume that the ground-state energy per site is given by

Truncate the above series at and find by solving the linear equation set numerically.