PROBLEM SET IX
“YOU’VE STRUCK OIL”
DUE FRIDAY,  NOVEMBER 
Exercise . Show that the function
{
x/(exp(x) − 1)
h(x) :=
1
if x ̸= 0;
if x = 0
is of class C∞ .
Exercise . For any integer m ≥ 0, denote by bm the value h(m) (0) of the m-th
derivative of h at 0. Show that b0 = 1, and for any integer m ≥ 1, show that bm is
the unique rational number so that
)
m (
∑
m+1
bk = 0,
k
k=0
(
)
m+1
(m + 1)!
=
k
(m + 1 − k)!k!
is the binomial coefficient.
Use this characterization to show that for any integer m ≥ 1, one has b2m+1 =
0 and b2m = (−1)m−1 |b2m |.
where
Exercise⋆ . Suppose now that p is any polynomial, and suppose that N ≥ 0
an integer such that for any integer k > N and any x ∈ R, one has p(k) (x) = 0.
Show that
∫ 1
N
∑
)
1
bk ( (k−1)
p = (p(0) + p(1)) −
p
(1) − p(k−1) (0) .
2
k!
0
k=2
Reflection: you have shown that integrals of polynomials can be computed by
the values of their derivatives at the endpoints! How can this be?
