MATH 101 ASSIGNMENT 5
1. In this assignment, we would like you to reflect on your experience in this course as a whole. As a team,
discuss the following prompts and write a one paragraph response to each.
(a) What are you most proud of accomplishing in this course, and why?
(b) Has your experience in this course changed your view of calculus as a subject? If so, how? If not,
why not?
(c) Write a brief message to the rest of the class to motivate them for the final exam. (Be creative!
The best messages will be given a bonus mark and may be posted anonymously on Piazza.)
∞
X
1
is known to diverge. In class, we proved this using the integral
n
n=1
test. Work through the steps below to build a proof that needs no calculus at all. Start with the
partial-sum notation
N
X
1
,
N ∈ N.
SN =
n
n=1
2. The famous Harmonic Series S =
(a) Evaluate S1 , S2 , S3 , and S4 .
1
(b) Explain why S2n+1 − S2n ≥ for each integer n ≥ 0.
2
Suggestion: Show your readers a pattern by expressing a few sample differences, like S2 −S1 , S4 −S2 ,
S8 − S4 , etc., in a natural format that makes the requested inequality easy to justify.
(c) Combine your findings in part (b) with the limit-definition for the value of S to explain why S = +∞.
3. Stacking boxes onto shelves at an Amazon warehouse is exhausting work. This problem may help you
sympathize with those who do it. In the usual good news/bad news dichotomy, the good news is that the
boxes here are two-dimensional, so they have no mass. The bad news is that there are infinitely many of
them. . . .
For each integer n ≥ 1, let an = 1/n2 be the area of a square of side length 1/n. (Call this “box number
n”.) Consider the task of piling all those boxes onto a horizontal shelf. Like physical boxes, one box can
rest on top of another, but they can’t occupy the same space.
Investigate the scheme where we put box number 1 on the left end of the shelf. Then we put down box
number 2 beside it on the right, and rest box number 3 on top of box 2. Box number 4 goes on the shelf
right beside box number 2, and we pile boxes 5, 6, 7 on top of box 4. The next box to land on the shelf
is number 8. This process continues. The boxes touching the shelf are numbered 1, 2, 4, 8, . . . , 2k , . . ..
(a) Make a reasonable sketch showing the positions of boxes 1, 2, . . . , 16, once the stacking process is
well underway.
(b) Imagine running the x-axis along the shelf, so that box 1 sits above the interval [0, 1]. Write a
formula1 involving n for the interval that supports box number 2n . (Call this interval [an , bn ].)
(c) Consider the pile of boxes that has box number 2n on the bottom. How many boxes are in this
pile? Explain carefully why the total height of this pile is not more than 1 unit.
1
(This should be valid for every integer n ≥ 0: you can check it for 0 ≤ n ≤ 4 by comparing the picture in (a).)
(d) Determine the total width of shelf required to hold the whole given sequence of boxes. Use it to
b such that the total area of all the boxes is not larger than A.
b
identify a specific number A
(e) The box-packing discussion above proves convergence for a certain series. Identify the series and
make intelligent remarks.
4. Let q > 0 be a real constant. Instead of using square boxes as in Question 3, imagine boxes in the shape
of rectangles, where box n has base width 1/nq and height 1/n. (Treat Question 3 as the special case
q = 1.)
(a) Imagine packing these new rectangular boxes according to the process detailed in Question 3. The
picture from 3(a) won’t change much, so do nothing here.
(b) Upgrade your formula from 3(b) to find the interval [an , bn ] supporting box number 2n . The result
should apply for any n ≥ 0 and any q > 0.
(c) Adapt your work in 3(c) to apply with a general q > 0. In particular, show that the height-limit of
1 remains correct.
(d) Express as a series the total width of shelf required to hold all the boxes. Then determine which
values of q > 0 correspond to a finite total width.
(e) The box-packing discussion above proves convergence for a certain family of series. Identify this
family and make intelligent remarks that relate your findings here with important results discussed
in class.