Indeterminate Hamburger moment problem: Entropy convergence

The indeterminate Hamburger moment problem is considered, jointly with all its real axis supported probability density functions. As a consequence of entropy functional concavity, out of such densities there is one which has largest entropy and that plays a fundamental role: we call it $f_{hmax}$. It is proved that the approximate Maximum Entropy (MaxEnt) densities constrained by an increasing number of moments converge in entropy to $f_{hmax}$ where the value of its entropy can be finite or $-\infty$.