ABSTRACT
It needs to be stressed that the error test has to be applied to consecutive error terms and not actualterms of the series. If it is merely known that q("'(0) and q("+"(0) are of opposite sign, then the relation
shows that (2.06) is certainly true for all x > X,, provided that X, is taken to be sufficiently large. But an actual value for X, is not available from this analysis. 2.3 When (q(")(r)l is not majorized by Iq'")(O)( we may consider the obvious extension
Very often, however, C, is infinite or else so large compared with (q(")(0)1 that this bound grossly overestimates the actual error. In these cases it is preferable to seek a majorant of the form
Iq(")(t)J < Iq(")(0)J een' (0 < r < co), (2.08) in which the quantity a, is independent of r. substitution of this majorant in (2.04) leads to
The best value of a, is evidently
Unlike (2.07) the ratio of the bound (2.09) to the actual value of I&,(x)l has the desirable property of tending to unity as x 4 co. The need to compute the derivatives of q ( t ) is sometimes a drawback. A later method ($9) avoids this difficulty. 2,4 As an illustration of the error bound of the preceding subsection, consider again the expansion of the incomplete Gamma function. If we set r = x(1 +T)
10 3 Integrals of a Real Variable
compare (2.04). From (2.10) we have
(a-l)(a-2) (a-n)e-"P-" Ien(x)l G (x > a-n-1 > 0). (2.13)
x - a + n + l
Moreover, it follows from (2.11) that e,(x) is positive in these circumstances. By comparison with $1.2 the bound (2.13) is a slightly weaker, but more concise result. For the particular case n = 0, we have
T (a, x) < - ( a > l , x>a -1 ) ; x - a+ l
compare (1.05). Ex. 2.1 Show that the sum or product of two alternating functions is itself alternating.
