ASSIGNMENT 2.2
There are two parts to this assignment. Part A is to be completed online before noon on Monday, January
24. Part B, which should be typeset (preferably with LaTeX), is to be submitted to the assignment box by the
same time.
Include your full name and student ID, along with the subject and assignment number, at the top of the front
page.
Part A [15 marks]
This part of the assignment can be found online, labelled Assignment 2.2, at www.mathxl.com. Questions
about the MathXL program itself should be sent to Eric; in fact, questions sent through the site will be directed
to him. To ask me questions, email me directly or come to my office.
Part B [15 marks]
1. Determine which of the following series converge.
X nn
(a)
3n n!
n≥1
X
X
(b)
ean , where
an converges
n≥1
(c)
X
n≥1
n≥1
X
an
, where
an converges absolutely
1 + an
n≥1
X (2n − 1)(2n − 3) · · · (5)(3)(1)
(d)
(2n)(2n − 2) · · · (4)(2)
n≥1
√
X n + 1 − √n
√
(e)
n
n≥1
2. Consider the sequence {an } defined by
1/n when n does not contain the digit 9
an =
.
0
otherwise
X
Determine whether
an converges.
n≥1
3. The Riemann zeta function is defined to be
ζ(s) =
X 1
.
ns
n≥1
(Here s is a real number greater than 1.) Show that
Y
ζ(s) =
primes p
1
,
1 − p−s
where the expression on the right-hand side is an infinite product taken over all primes p:
Y
1
1
1
1
=
··· .
1 − p−s
1 − 2−s
1 − 3−s
1 − 5−s
primes p