Concrete mathematics : a foundation for computer science

138 ASYMPTOTICS
Table 438 Asymptotic approximations, valid as n + 00 and z + 0.
5- H, = lnn+y+&-A+& +O (‘). 2
?A!-
(9.28)
. (9.29)
B, = 2[n even](-1 )n,/2 ’ &(l+2pn+3~n+O(4mn)). (9.30)
-4 n(n) = & + ilntj2 + 2!n
-+&$+o(&& (Inni
(9.31)
ez = ‘+r+;+~+~+o(r5i. (9.32)
ln(l+z) = z-f+$-~+0(z5). (9.33)
1
~ = 1 +z+z2+23+t4+0(25).
1-z
(9.34)
(1 +z)a = 1 +cxz+ (;)d+ (;)z3+ (;)24+o(z’l (9.35)
An asymptotic approximation is said to have absolute error 0( g(n)) if
it has the form f(n)+O(g(n)) w h ere f(n) doesn’t involve 0. The approximation
has relative error O(g(n)) if it has the form f(n)(l + O(g(n))) where
f(n) doesn’t involve 0. For example, the approximation for H, in Table 438
has absolute error O(n 6); the approximation for n! has relative error O(n4).
(The right-hand side of (9.29) doesn’t actually have the required form f(n) x
(1 + O(n “)), but we could rewrite it
dGi (f)n(l + & + & - ‘) (1 + O(nP4))
5 1 840n3
if we wanted to; a similar calculation is the subject of exercise 12.) The (Relative error
absolute error of this approximation is O(n” 3.5e ~-“). Absolute error is related
to the number of correct decimal digits to the right of the decimal point if
is nice for taking
reciprocals, because
,,(, + 0(c)) =
the 0 term is ignored; relative error corresponds to the number of correct
“significant figures!’
We can use truncation of power series to prove the general laws
1 +0(E).)
ln(l + O(f(n))) = O(f(n)) , if f(n) < 1; (9.36)
e”‘f’n)l = 1 +O(f(n)) , if f(n) = O(1). (9.37)