ASSIGNMENT 9
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 by 3:00 on Thursday, April 2. The online assignment will
close at 9:00 a.m. on Friday, April 10.
1. (a) Write down the Taylor series for sin(x) centred at
writing down the first few terms.)
π
3.
(You may do this using sigma notation or by
(b) Prove that this Taylor series converges to sin(x) for all x.
(
2. Let f (x) =
e−1/x
0
2
if x 6= 0
.
if x = 0
(a) Prove that f (x) is infinitely differentiable at 0. (It should be clear that it is infinitely differentiable
at all x 6= 0.)
(b) Explain why the Taylor series for f (x) centred at 0 (i.e. its Maclaurin series) does not converge to
f (x) on any open interval.
3. (a) Calculate
x3
x→0 x − tan(x)
lim
in two ways: (i) by using an appropriate power series, and (ii) by using l’Hospital’s Rule.
(b) Compare the two calculations in part (a). Which is more attractive? Defend your answer in a few
sentences.
4. [Bonus] Compare the course content of MATH 100 and MATH 101. Which is more attractive? Defend
your answer in a few sentences.