ASSIGNMENT 7
There are two parts to this assignment. The first part consists of questions 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 justifications. You will be graded on the mathematical, logical and
grammatical coherence and elegance of your solutions. Your solutions must be typed, with your name and
student number at the top of the first page. If your solutions are on multiple pages, the pages must be stapled
together.
Your written assignment must be handed in before the start of your recitation on Friday, March 20.
The online assignment will close at 9:00 a.m. on Friday, March 20.
1. Does
X (−1)n n2 2n
converge? Justify your answer.
n!
n≥1
2. The alternating harmonic series
X (−1)n−1
n
n≥1
converges conditionally: it converges (by the Alternating Series Test), but the series
X (−1)n−1 n
n≥1
diverges. Explain how the terms in the alternating harmonic series can be rearranged to converge to π.
3. The Riemann zeta function
ζ(s) =
X 1
ns
n≥1
is of central importance in the field of analytic number theory. In this question you will prove that
ζ(s) =
1
1
1
−
ps
primes p
Y
(1)
for real numbers s > 1. The symbol on the right-hand side denotes a product — in this case, taken over
all primes p.
X
X
(a) Let
an and
bn be positive series which converge. Prove that their product
n≥1
n≥1


X

an  
n≥1

X
n≥1
bn  =
n
XX
ak bn−k+1
n≥1 k=1
converges. (Hint: prove that the partial sums form a bounded, monotone sequence.)
X 1
Y
1
(b) Explain why
> ζ(s) −
.
s
n
1 − p1s
primes
n≥q+1
p≤q
(c) Explain in a few sentences why the identity (1) is justified.