ASSIGNMENT 14
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. You are expected to provide full solutions
with complete arguments and justifications. You will be graded on the correctness, clarity and elegance of
your solutions. Your answers must be typed or very neatly written. They must be stapled, with your name
and student number at the top of each page.
This assignment is due at 10:00 a.m. on Friday, March 14. The written assignment is to be handed in
by that time. The online assignment will also close at that time.
1. Determine whether each of the following series converges.
X
X
an converges and an ≥ 0 for all n
a2n , where
(a)
n≥1
n≥1
(b)
X
n
(−1) an , where an =
n≥1
1
n
1
n2
if n is odd
if n is even
X 22n (n!)2
(2n)!
n≥1
√
√ X
n+1
n
−
(d)
n+3
n+4
(c)
n≥1
2. The alternating harmonic series
X (−1)n−1
n
n≥1
converges; this can be shown using the Alternating Series Test. In particular, it converges conditionally;
that is, it converges but
X (−1)n−1 n
n≥1
diverges. It turns out that one can rearrange the terms in any conditionally convergent series so that
the rearranged series converges to anything one chooses, or diverges however one chooses. Prove that the
terms of the alternating harmonic series may be rearranged so that the rearranged series converges to π.
3. We define the infinite product
Y
an to be the formal product of elements a1 , a2 , a3 , · · · . (Convergence
n≥1
is defined analogously to convergence of series, by considering partial products.) Justify in one or two
paragraphs the identity
Y
X 1
1
=
.
−s
1−p
ns
all primes p
n≥1
You may assume if you wish that s is a real number greater than 1.