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 Limit Theorems for Sequences

READING §9

\underline{Warm-Up} (5-10 minutes)

USEFUL THEOREMS Now that we know how to formally prove that a sequence convergence to a particular limit, we know that we don’t want to use this formal method everytime we want to find the limit of a sequence! This is why people develop theorems based on simpler results. We will be looking at Theorems 9.1, 9.2 in depth and leave the rest of the chapter up to you. \underline{Theorem:} Convergent series are bounded

\underline{Theorem:} lim(ks_{n}) = k lim s_{n} If the sequence (s_{n}) converges to s and k ∈ ℝ, then the sequences (ks_{n}) converges to ks. That is, lim(ks_{n}) = k lim s_{n}.

\underline{Theorem:} lim(s_{n} + t_{n}) = lim s_{n} + lim t_{n} If the sequence (s_{n}) converges to s and (t_{n}) converges to t, then (s_{n} + t_{n}) converges to s + t.

\underline{Theorem:} lim(s_{n}t_{n}) = (lim s_{n})(lim t_{n}) If the sequence (s_{n}) converges to s and (t_{n}) converges to t, then (s_{n}t_{n}) converges to st.

\underline{Theorem:} lim(1/s_{n}) = 1/(lim s_{n}) If the sequence (s_{n}) converges to s and s_{n} ≠ 0∀n, and if s ≠ 0 then (1/s_{n}) converges to s, then (s_{n}t_{n}) converges to st.

Homework Assignment §8, page 32: 9.1(b), 9.2(b), 9.5 and 9.6