Concrete mathematics : a foundation for computer science

Asymptotics is
the art of knowing
where to be sloppy
and where to be
precise.
9.4 TWO ASYMPTOTIC TRICKS 153
3 Now prove that
each of the following three sums is small:
L(n) =
xc
x ok(n); tb(n) = x ‘Jk(n) ;
MT, kET,
(n) = x (ck(n)l. (9W
If all three steps can be completed successfully, we have a good estimate:
t ak(n) = t bk(n) + o(L(n)) + O(xb(n)) + o(L(n)) .
kED,uT, kED,uT,
Here’s why. We can “chop off” the tail of the given sum, getting a good
estimate in the range D, where a good estimate is necessary:
x ak(n) = x @k(n) + O(ck(n))) = t bk(n) + o&(n)).
G-D, kCD, ND,
And we can replace the tail with another one, even though the new tail might
be a terrible approximation to the old, because the tails don’t really matter:
x ak(n) = x @k(n) - bk(n) + ak(n))
&T, MT,
= x h(n) + O(xb(n)) + o&,(n)).
MT,
When we evaluated the sum in (g.6o), for example, we had
ak(n) = [06k