Concrete mathematics : a foundation for computer science

In other words, we need to have
9.5 EULER’S SUMMATION FORMULA 459
(-l)mBm = B,,(l) = B,(O), for m > 1. (9.75)
This is a bit embarrassing, because B,(O) is obviously equal to B,, not
to (-l)mB,. But there’s no problem really, because m > 1; we know that
B, is zero when m is odd. (Still, that was a close call.)
To complete the proof of Euler’s summation formula we need to show
that B,,,(l) = B,(O), which is the same as saying that
for m > 1.
But this is just the definition of Bernoulli numbers, (6.7g), so we’re done.
The identity B&(x) = mBm-l (x) implies that
1
s 0
Bm(x) dx = B ,+1(l) - Bm+l(O)
m+l
and we know now that this integral is zero when m 3 1. Hence the remainder
term in Euler’s formula,
R, = (-‘);+’ i” Bm((x))f(“‘)(x) dx,
m. a
multiplies f’“)(x) by a function B, ({x}) whose average value is zero. This
means that R, has a reasonable chance of being small.
Let’s look more closely at B,(x) for 0 6 x 6 1, since B,(x) governs the
behavior of R,. Here are the graphs for B,(x) for the first twelve values of m:
Bm(x)
m := 1 m=2 m=3
/
,
W-
B 4+m(X) - - -
BS+m(X) - - -
m=4
24
Although BJ (x) through Bg(x) are quite small, the Bernoulli polynomials
and numbers ultimately get quite large. Fortunately R, has a compensating
factor 1 /m!, which helps to calm things down.