ASSIGNMENT 15
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 Monday, March 24. The written assignment is to be handed
in by that time. The online assignment will also close at that time.
1. Recall that a number is said to be transcendental if it is not algebraic; that is, if it is not a root of
a polynomial with integer coefficients. We have seen that there are uncountably infinite real numbers,
and it may be shown that the number of algebraic numbers is countable; thus there are uncountably
infinite transcendental numbers. Showing that a particular number is transcendental is considerably
more challenging. In this problem, you will do just that.
(a) Let α be an irrational number which is the root of a degree-n polynomial f with integer
coefficients.
Let s be the maximum value of |f 0 (x)| on the interval (α−1, α+1). Finally, let A = min 1, 1s , |α − r| ,
where r is the root of f which is nearest α (but not α itself).
Let pq be a rational number satisfying α − pq ≤ qAn . Prove that this is impossible, by proving that it
implies α − pq > qAn .
(Hint: apply the Mean Value Theorem to f on the interval between α and pq , and solve for α − pq .)
(b) Let α =
m
X 1
X
1
m!
,
p
=
10
and q = 10m! . Prove that there are values of m such that
10k!
10k!
k≥1
α − pq ≤ qAn .
k=1
(Hint: write and evaluate α − pq as a geometric series.)
(c) Explain in a few sentences why you may conclude that α =
X 1
is transcendental.
10k!
k≥1
2. Evaluate the following.
X x2n
, at the left endpoint of its interval of convergence.
3n
n≥1
X n
(b)
.
3n
(a)
n≥1
3. (a) Find the Maclaurin series, and radius of convergence, for arctan x.
(b) Use your answer in part (a) to find a series expression for π. (Note that this provides one method of
calculating π to any degree of accuracy. Your answer to Question 2 on Assignment 14 may seem to,
but it requires you to know a priori the value of π.)