On Stirling’s formula remainder

Let the sequence r n be defined by n ! = 2 π n n / e n e r n . We establish new estimates for Stirling's formula remainder r n . This improves some known results. Let ? ( x ) be defined by the relation: ? ( x + 1 ) = 2 π ( x / e ) x e ? ( x ) / ( 12 x ) . We prove that ? ? ( x ) is completely monotonic on ( 0 , ∞ ) . This implies the result given by Mortici, who proved that - x - 1 ? ? ( x ) is completely monotonic on ( 0 , ∞ ) .