Math 220, Sessional Examinations December 2001
1. (12 points)
A. Compute the first 4 terms of the sequence (xn ) defined by
n+2
xn + 1, n ≥ 1.
n
B. Guess a formula for xn and prove that your formula is correct.
x1 = 1, xn+1 =
2. (12 points)
A. Give, with explanation, an example of an infinite bounded subset of the real numbers that has no interior points.
B. Give an example of a subset of the real numbers that has exactly
220 interior points, or show that no such set exists.
3. (15 points)
√
A. Define a sequence by x1 = 3 and xn+1 = 6xn − 8 for n ≥ 1.
Prove that (xn ) converges and find limn→∞
√ xn .
B. If, instead, we set x1 = a and xn+1 = 6xn − 8 for n ≥ 1,
where a is some real number, for what values of a does (xn )
converge? Prove your answer.
4. (12 points) Find the limit
2n + 1
.
n→∞ 2n − 1
lim
Use the definition of limit (i.e. with and N) to prove your answer.
5. (12 points) Determine, with justification, whether the following
series converge absolutely, converge conditionally, or diverge.
A.
∞
X
(−1)n π n n
22n
n=0
B.
∞
X
(−1)n
√
n− n
n=2
6. (12 points)
A. Use the definition of continuity (i.e. with and δ) to show that
f (x) = x2 − 5x + 2 is continuous at x = 3.
B. Let g(x) = x5 +2x3 +4. Find an integer n such that there exists
a real number x0 ∈ [n, n + 1] with g(x0 ) = 0. Fully justify your
answer (you do not have to use − δ proofs!).
7. (15 points) Determine whether the following statements are true
or false. Fully justify your answers.
A. There exists a subset S ⊆ R whose closure is Q.
B. The set S = n12 : n ∈ N is compact.
1
2
C. If (an ) is a sequence for which
lim a2n = lim a2n+1 = L ∈ R,
n→∞
then limn→∞ an = L.
8. (10 points) Define
an =
n→∞
1
1
n
if n is even
if n is odd
Find, with proof, the radius of convergence of the power series
P∞
n=1
an xn .