ABSTRACT
If p > I , then t ( ' l f l ) - l -+ co. No problem arises if a,,,, and a,, have opposite signs because the right-hand side tends to - cu as t -+ 0. But ifp > 1 and a,, ,/a, is positive, then a,= a. 9.2 A simple way of overcoming the difficulty is to modify the majorant (9.01) by the inclusion of an arbitrary factor M exceeding unity; M = 2 would be a realistic choice in many circumstances. Then in place of (9.02) we derive
where
Then an = 0, which means that the ratio of the bound (9.04) to the absolute value of the first neglected term in the asymptotic expansion is M,, independently of x. In practice, however, M,, may turn out to be infinite or unacceptably large. 9.3 Another approach is to let m be the largest integer such that m < p , and a,,+,+ , the first member of the set a,,+ ,, an+ , . . ., a,, +, that has opposite sign to an, or, if no such member exists, let j = m. Define
The advantage of (9.07) is that its ratio to the absolute value of the actual error tends to unity as x -r oo, unlike (9.04). Disadvantages are increased complexity and the need to evaluate coefficients beyond a,. 9.4 In the case of Theorem 8.1, it is seen from the proof that the nth truncation error of the expansion (8.08) can be expressed
In other cases, we have from (1.05) and (2.14)
Substituting in (8.1 1) by means of this inequality, we obtain
where K = p(k)-p(a) as before, and a, = max {(n+rl-I(- l)/p,O). (9.10)
The second error term, e-xP(a)~n,2(x), can be bounded by methods similar to those of s9.1 to 9.3. The role of r is now played by v, and +,(t) is replaced by ~(~+'-")lrfn(v); the essential difference is that the suprema in (9.03). (9.05), and (9.06) are evaluated over the range 0 < v < K instead of 0 < r < oo. The bounds (9.02), (9.04), and (9.07) apply unchanged to (x)l.
