Sequences that converge quickly to a generalized Euler constant

The purpose of this paper is to give some sequences that converge quickly to a generalized Euler constant, i.e. the limit of the sequence ( 1 a s + 1 ( a + 1 ) s + ⋯ + 1 ( a + n − 1 ) s − 1 1 − s ( ( a + n − 1 ) 1 − s − a 1 − s ) ) n ∈ N , where a ∈ ( 0 , + ∞ ) and s ∈ ( 0 , 1 ) .