SCIENCE ONE: MATHEMATICS ASSIGNMENT 13 (Due Mar. 16, 2012)
There are two parts to this assignment. The first part is online and is due by 12:00 noon on Fri. Mar. 16;
you will need to login at www.mathxl.com. The second part consists of the questions on this page and is also
due by 12:00 noon on Fri. Mar. 16. 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 the first page.
1. Determine (with proof) whether
P∞
k+1
ak
k=1 (−1)
(
ak =
converges or diverges, if
10/k 2
if k is even
3
1/(10k )
if k is odd.
2. Find a power series representation for ln(1 + 2x) centred at 0, and determine (with proof) the interval of
convergence for the power series. Hint: apply Theorems 9.4 and 9.5 carefully.
3. Find the sum of
∞
X
(−1)k k
x2
x3
x4
x5
x =
−
+
−
+ ···
k(k − 1)
2
3·2 4·3 5·4
for |x| < 1,
k=2
and determine (with proof) the interval of convergence for the power series. Hints: differentiate, apply
Theorem 9.5 carefully.