MATHEMATICS: ASSIGNMENT 8
There are two parts to this assignment: Part A and Part B. Part A is to be completed online, and Part B
is to be handed in, both before 10:00 a.m. on Friday, January 29.
Part A
This part of the assignment will be found online, labelled Assignment8A, at webwork.elearning.ubc.ca.
Part B
1. Consider the sequence
an =
n
X
1
i=1
i
!
− log n.
Prove that {an } converges. (Hint: show that {an } is bounded and monotonic.) The limit, denoted γ, is
called the Euler-Mascheroni constant.
2. (a) Prove that the alternating harmonic series
X (−1)n−1
converges.
n
n≥1
2. (b) Show (by induction, or otherwise) that
2n
X
i=1
2n
n
X1 X1
(−1)i−1
=
−
.
i
i i=1 i
i=1
3. Conclude, using questions 1 and 2, that the alternating harmonic series converges to log 2.
4. Determine whether
XX
m≥1 n≥1
Z
5. Let F (x) =
x
sin
0
1
converges.
(m + n)2
1
1
dt. Find F 0 (0). (Hint: integrate t cos
using integration by parts.)
t
t