SCIE 001 MATHEMATICS ASSIGNMENT 12 (Due Mar. 15, 2013 )
There are two parts to this assignment. The first part is on WeBWorK and is due by 10:00 am on Wed.
Mar. 15. The second part consists of the questions on this page, and is due before 10:00 am on Wed. Mar.
15. For these questions, you are expected to provide full solutions with complete arguments and justifications.
You will be graded primarily on the correctness, clarity and elegance of your solutions. Your answers must
be typeset or very neatly written. They must be stapled, with your name and student number at the top of
each page.
1. Recall in class we made the following statements: i) the sequence
1 ∞
3n−1
n=1
(−1)n n
2n+1
o∞
is convergent, and ii) the sequence
n
n=1
is divergent.
(a) Justify both statements i) and ii) using the “informal” Definition 1, p. 692.
(b) Prove both statements i) and ii) using the “precise” Definition 2, p. 692.
2. Do Problem 5, p. 782 (find the length and enclosed area of the snowflake curve).
3. In this problem, you prove that the “alternating harmonic series”
∞
X
(−1)n−1
n
n=1
is convergent, by proving that the sequence of partial sums
sn =
n
X
(−1)i−1
i
i=1
is convergent.
(a) Prove that the sequence of even partial sums {s2m } is increasing and bounded above. Therefore, by
the Monotone Sequence Theorem,
lim s2m = s
m→∞
exists (as a finite number).
(b) Observing that the odd partial sums can be expressed as
s2m+1 = s2m +
1
2m+1 ,
show that
lim s2m+1 = s,
m→∞
the same value s as in part (a).
(c) Prove that the entire sequence of partial sums is convergent,
lim sn = s,
n→∞
using Definition 2, p. 692.