Spring 2005 MATH 172
Week in Review VII
courtesy of David J. Manuel
Section 10.1
1. State the precise definition for each of the following statements:
a) lim an = L
n→∞
b) lim an = ∞
n→∞
3
=0
n→∞ n + 3
2. Use the definition to prove lim
3. Use the definition to prove lim 3−n = 0.
n→∞
4. Prove the following: If an → L1 and bn → L2 , then an + bn → L1 + L2 .
5. State and prove the Squeeze Theorem for sequences.
6. State the Monotone Sequence Theorem. Use it to prove that lim
n→∞
ln n
exists. Find the limit.
n
1
7. Use induction to prove that the sequence defined recursively by a0 = 1, an+1 = an + 1 is increas2
ing and 1 ≤ an < 3 for all n. Explain why the limit exists, then find it.
2n+1 − 1
8. Use induction to prove that the sequence defined in #7 can be written as an =
for all
2n
n.