Concrete mathematics : a foundation for computer science

Remember that
R stands for “‘real
part.”
9.3 0 MANIPULATION 437
Power series give us some of the most useful operations of all. If the sum
S(z) = tanz”
n>O
converges absolutely for some complex number z = a, then
S(z) = O(l),
This is obvious, because
for all 121 6 /22/.
In particular, S(z) =: O(1) as z + 0, and S(l/n) = O(1) as n + 00, provided
only that S(z) converges for at least one nonzero value of z. We can use this
principle to truncate a power series at any convenient point and estimate the
remainder with 0. For example, not only is S(z) = 0( 1 ), but
S(z) = a0 +0(z),
S(z) = a0 + al2 + O(z2) ,
and so on, because
S(z) = x ukzk +zm x a,znem
O$km
and the latter sum is 0 (1). Table 438 lists some of the most useful asymptotic
formulas, half of which are simply based on truncation of power series
according to this rule.
Dirichlet series, which are sums of the form tka, ak/k’, can be truncated
in a similar way: If a Dirichlet series converges absolutely when z = a, we
can truncate it at any term and get the approximation
t ok/k’ + O(m-‘) ,
l