The ring of power sums is formed by complex functions on ℕ of the form α ( n ) = b 1 c 1 n + b 2 c 2 n + ... + b h c h n , for some b i ∈ ℚ ¯ and c i ∈ ℤ . Let F ( x , y ) ∈ ℚ ¯ [ x , y ] be absolutely irreducible, monic and of degree at least 2 in y . We consider Diophantine inequalities of the form | F ( α ( n ) , y ) | < | ∂ F ∂ y ( α ( n ) , y ) | · | α ( n ) | - ε and show that all the solutions ( n , y ) ∈ ℕ × ℤ have y parametrized by some power sums in a finite set. As a consequence, we prove that the equation F ( α ( n ) , y ) = f ( n ) , with f ∈ ℤ [ x ] not constant, F monic in y and α not constant, has only finitely many solutions.