Concrete mathematics : a foundation for computer science

464 ASYMPTOTICS
The integral JT B4({x})f’“:(x) dx will exist whenever f’“‘(x) = 0(x ‘) as
x + 00, and in this case f14) (x) surely qualifies. Moreover, we have
R4(n) = Rl(m!+~~Bl({x:)(~-~)dn
= R4(00) + O(/“xp6 dx ) = R~(cw) + O(nP5).
n
Thus we have used Euler’s summation formula to prove that
= ln2 -t s -n-l + R4(00) + O(n5)
= C - I-I-’ + O(ne5)
for some constant C. We do not know what the constant is-some other
method must be used to establish it -but Euler’s summation formula is able
to let us deduce that the co,nstant exists.
Suppose we had chosen a much larger value of m. Then the same reasoning
would tell us that
R,(n) = R,(m) + O(nPmP’),
and we would have the formula
1
~ =
t
C - T-C’ + c2nP2 + cjnP3 + . . . + c,n~~ m + O(nPmP’ )
1 , 1 . (9.82)
This is not enough to prove that the sum is exactly equal to C - n ’ ; the
actual value may be C - n’ + 2-” or something. But Euler’s summation