Basic asymptotic estimates for powers of Wallis’ ratios

For any a ∈ R , for every n ∈ N , and for n -th Wallis’ ratio w n := (cid:81) nk =1 2 k − 1 2 k , the relative error r 0 ( a, n ) := (cid:0) v 0 ( a, n ) − w an (cid:1) /w an of the approximation w an ≈ v 0 ( a, n ) := ( πn ) − a/ 2 is estimated as (cid:12)(cid:12) r 0 ( a, n ) (cid:12)(cid:12) < 14 n . The improvement w an ≈ v ( a, n ) := ( πn ) − a/ 2 (cid:16) 1 − a 8 n + a 2 128 n 2 (cid:17) is also studied.