When the remainder is bounded in magnitude by the first neglected term, and has the same sign when . When the remainder term is bounded in magnitude by times the first neglected term. For these and other error bounds see Olver (1997b, pp. 109–112) with .
If the expansion is terminated at the th term, then the remainder term is bounded by times the next term. For the function see §9.7(i).
as in .
The remainder terms are given by
where, for ,
When , these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when . When the remainders are bounded in magnitude by times the first neglected terms.