ASSIGNMENT 6
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 13.
The online assignment will close at 9:00 a.m. on Friday, March 13.
1. For what values of k does
X log(n)
converge?
n3k
n≥1
2. Imagine you have an infinite number of cubes, of side lengths 1, 21 , 13 , 41 , . . .. Prove that these can all fit
inside a rectangular box of volume 32 .
3. (a) Let
an =
Prove that
X
1
n
0
if n does not contain the digit “9”
.
if n contains the digit “9”
an converges.
n≥1
(b) Let
bn =
Do you think
X
1
n
0
if n does not contain the digit string “1729”
.
if n contains the digit string “1729”
bn converges? Justify your answer. (A full proof is not expected, but it is certainly
n≥1
one way of getting full marks.)