Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

(09/20/96)

SUMMARY More Limit Theorems and Infinite limits READING §9 \underline{Warm-Up} (5-10 minutes)

MORE USEFUL THEOREMS We can use the following theorems to find the limits of typical sequences.

(a) lim_{n→∞}(1/n^{p}) = 0 for p > 0

(b) lim_{n→∞}(a^{n}) = 0 if |a| < 1

(c) lim(n^{1/n}) = 1

(d) lim_{n→∞}(a^{1/n}) = 1 for a > 0 \underline{Examples}

Find lim \frac{n − 5}{n^{2} + 6}

Find lim \frac{n^{2} + 6}{n − 5}

\underline{Definition:} Infinite limits For a sequence (s_{n}) we say that lim s_{n} = +∞ provided that

∀M > 0 ∃N ∋ n > N ⇒ s_{n} > M

(write this out in your own words)

and we say that lim s_{n} = −∞ provided that

∀M < 0 ∃N ∋ n > N ⇒ s_{n} < M

(write this out in your own words)

We say that lim s_{n} exists if

• (s_{n}) converges

• (s_{n}) diverges to +∞ or −∞

\underline{Examples} Prove lim_{n→∞}(\sqrt{n} − 9) = +∞

Prove lim_{n→∞} \frac{n^{3} − 4}{n^{2} − 2} = +∞

Homework Assignment (Due 9/30/96) §9, page 40-41: 9.1(b), 9.2(b), 9.5, 9.6, 9.7, 9.8, 9.13, 9.15, 9.16, 9.17