New upper bounds of n!

In this article, we deduce a new family of upper bounds of n ! of the form n ! < 2 π n ( n / e ) n e M n [ m ] n ∈ â„• , M n [ m ] = 1 2 m + 3 1 4 n + ∑ k = 1 m 2 m - 2 k + 2 2 k + 1 2 - 2 k ζ ( 2 k , n + 1 / 2 ) m = 1 , 2 , 3 , . . In this article, we deduce a new family of upper bounds of n ! of the form n ! < 2 π n ( n / e ) n e M n [ m ] n ∈ â„• , M n [ m ] = 1 2 m + 3 1 4 n + ∑ k = 1 m 2 m - 2 k + 2 2 k + 1 2 - 2 k ζ ( 2 k , n + 1 / 2 ) m = 1 , 2 , 3 , . .