Concrete mathematics : a foundation for computer science

460 ASYMPTOTICS
The graph of B,(x) begins to look very much like a sine wave when
m > 3; exercise 58 proves that B,(x) can in fact be well approximated by a
negative multiple of cos(27rx - inm), with relative error l/2”.
In general, Bdk+l (x) is negative for 0 < x < i and positive for i < x < 1.
Therefore its integral, Bdk+~ (x)/(4k+2), decreases for 0 < x < 5 and increases
for i < x < 1. Moreover, we have
bk+l(l - X) = -&+I (X) ,
and it follows that
bk+2(1 -X) = bk+2(x),
for 0 < x < 1,
for 0 < x < 1.
The constant term Bdk+2 causes the integral sd l&k+l(x) dx to be zero; hence
B4k+2 > 0. The integral of Bak+Z(X) is Bdk+A(x)/(4k+3), which must therefore
be positive when 0 < x < 5 and negative when i < x < 1; furthermore
B4k+3 ( 1 - x) = -B4k+3 (x) , so B4k+j (x) has the properties stated for i&k+1 (x),
but negated. Therefore B4k +4(x) has the properties stated for BJ~+z(x), but
negated. Therefore B4k+s(x) has the properties stated for B4k+l (x); we have
completed a cycle that establishes the stated properties inductively for all k.
According to this analysis, the maximum value of Blm(x) must occur
either at x = 0 or at x = i. Exercise 17 proves that
BZm(;) = (21mm2”’ - 1)‘B2,,,; (9.76)
hence we have
(bn(b4I 6 IBzml. (9.77)
This can be used to establish. a useful upper bound on the remainder in Euler’s
summation formula, because we know from (6.89) that
IBZllJ
(2m)!
Therefore we can rewrite Euler’s formula (9.67) as follows:
x f(k) = J”
a