MATH 295 PROBLEMS 3

IGOR KRIZ

1. Calculate

lim_{n→∞} [\frac{1 + 3 + 5 + 7 + ... + (2n − 1)}{n + 1} − \frac{2n + 1}{2}] .

[Hint: find a formula for 1 + 3 + ... + 2n − 1.]

2. Calculate

lim_{n→∞} \frac{n^{3} − 1}{(n − 1)^{3}} .

3. Let 0 < a_{1} < b_{1} and define

a_{n+1} = \sqrt{a_{n}b_{n}}, b_{n+1} = \frac{a_{n} + b_{n}}{2} .

Prove that the sequences a_{n} and b_{n} each converge, and have the same limit. [Hint: Prove first that for any two positive numbers a, b, \sqrt{ab} ≤ \frac{a+b}{2}.]

4. Suppose that

lim_{n→∞} a_{n} = 0

for a sequence (a_{n}) where a_{n} > 0 for each positive integer n. Prove that the set of all the numbers a_{n}, n ∈ ℕ actually has a maximum element.

5. Consider the sequence

\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \frac{1}{6}, ... .

For which numbers α is there a subsequence converging to α? [Recall that a subsequence of (a_{n}) is a sequence of the form (a_{i_{n}}) where i_{1} < i_{2} < ... < i_{n} < ....]