MATHEMATICS: ASSIGNMENT 7
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 22.
Part A
This part of the assignment can be found online, labelled Assignment7A, at webwork.elearning.ubc.ca.
Part B
1. Prove the Root Test:
If lim |an |
1/n
< 1, then
n→∞
X
|an | converges. If lim |an |
n→∞
n≥1
1/n
> 1, then
X
|an | diverges.
n≥1
2. Determine whether the following series converge:
X
X
2. (a)
ean , where
an converges
n≥1
2. (b)
X
n≥1
2. (c)
n≥1
X
an
, where
|an | converges (and an 6= −1 for all n)
1 + an
X
n≥1
n≥1
n
n+1
n2
3. A series is said to be conditionally convergent if it converges but does not converge absolutely. Show that
the terms of any conditionally convergent series may be rearranged so that the series converges to any
given number L.
4. 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
5. 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