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 the largest entropy density. It is proved that the approximate Maximum Entropy (MaxEnt) densities constrained by an increasing number of moments converge in entropy to the largest entropy density where the value of its entropy can be finite or −∞.