PROBLEM SET XI
“TAKE IT TO THE …”
DUE TUESDAY,  NOVEMBER
Exercise . Suppose (an )n≥0 and (bn )n≥0 two sequences of positive real numbers. Which
of the following is greater? In each case, give examples when equality obtains, as well as
examples when it does not.
lim sup(an + bn ) and lim sup an + lim sup bn ;
n→∞
n→∞
n→∞
(
)(
)
lim sup(an bn ) and
lim sup an
lim sup bn ;
n→∞
lim sup a−1
n
n→∞
n→∞
n→∞
(
)−1
and
lim sup an
;
n→∞
lim inf(an + bn ) and lim inf an + lim inf bn ;
n→∞
n→∞
n→∞
(
)(
)
lim inf(an bn ) and
lim inf an lim inf bn ;
n→∞
n→∞
n→∞
(
)−1
.
lim inf a−1
and
lim inf an
n
n→∞
n→∞
∑
Exercise . Suppose
∑ 2 (an )n≥0 a sequence of positive real numbers. Prove that an converges only if an converges. Give an example to show that the positivity condition is
necessary.
∑
Exercise . Suppose (an )n≥0 a sequence of positive real numbers. Prove that if an
converges, then
∑ 1
1 + an
diverges, and
∑ an
1 + an
converges.
Exercise⋆ . Consider the function w : [0, 1] . R given by
{
x−x if x ̸= 0;
w(x) :=
1
if x = 0.
Prove that w is continuous, prove that the series
∞
∑
m−m
m=1


DUE TUESDAY,  NOVEMBER
converges, and show that
∫
1
w=
0
∞
∑
m−m .
m=1
Definition. Suppose (an )n≥0 a sequence of complex numbers. en the infinite product
∞
∏
an
n=0
is the sequence
(
N
∏
)
n=0
N≥0
of partial products. We say that the infinite product conerges just in case this sequence of
partial products converges.
Exercise∏
. If (an )n≥0 is a sequence of positive real numbers, show that the infinite
product ∞
n=0 an converges just in case the series
∞
∑
log an
n=0
converges.
Exercise . Suppose s a positive real number. Show that the infinite product
∞
∏
(1 + 1/n)s
G(s) :=
(1 + s/n)
n=1
converges, and prove that G(1 + s) = (1 + s)G(s). Conclude that for any positive integer
m, one has m! = G(m).
Definition. If k is a positive integer, then the generalized harmonic numbers Hn (k) are
the partial sums of the harmonic series:
n
∑
1
Hn (k) :=
.
mk
m=1
Exercise . Consider the sequence (xn )n≥1 with
xn := Hn (1) − log n.
Prove that this sequence converges, even though neither (Hn (1))n≥1 nor (log n)n≥1 converges. Write γ := limn→∞ xn . is is called the Euler–Mascheroni constant.
Exercise⋆⋆ . Show that the function G : (0, +∞) . R defined above is differentiable
(in fact it’s C∞ !), and prove that for any nonnegative integer n, one has
G ′ (1 + n) = n! + (n + 1)!(Hn (1) − γ).
Take a moment to reflect on the madness of this formula. (To show that G is differentiable,
try to write a formula for G ′ (s)/G(s). To compute the value G ′ (1 + n), use induction to
reduce to the case of n = 0.)
PROBLEM SET XI
“TAKE IT TO THE …”

Definition. Suppose k an integer. en the k-th polylogarithm is the series
∞
∑
1 n
Lik (z) :=
z .
nk
n=1
Exercise . Compute the radius of convergence of Lik , and note that it is at least 1.
Exercise . Show that if k is a negative integer, then (z − 1)1−k Lik (z) is a polynomial
of degree −k with integral coefficients. What are the roots of this polynomial? What can
you say about its coefficients?
Exercise . Compute the radius of convergence of the power series
∞
∑
Mk (z) :=
Hn (k)zn .
n=1
Show, moreover, that within this radius of convergence, one has
(z − 1)Mk (z) = Lik (z).