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

#20 Monday (10/28/96)

SUMMARY Finishing Infinite Series, Continuity of Functions

READING §14-15 (pages 70-79)

Cauchy Criterion The series ∑ a_{n} satisfies the Cauchy Criterion if its sequence of partial sums (s_{n}) is a Cauchy sequence:

∀ϵ > 0∃N ∋ m, n > N ⇒ |s_{n} − s_{m}| < ϵ

If we assume that n > m and that we can use m − 1 in place of m then this becomes:

∀ϵ > 0∃N ∋ m − 1, n > N ⇒ |s_{n} − s_{m−1}| < ϵ

which is equivalent to:

∀ϵ > 0∃N ∋ m − 1, n > N ⇒ | ∑_{k=m}^{n} s_{k}| < ϵ

\underline{Theorem} (14.4 ∑ a_{n} converges if and only if it satisfies the Cauchy Criterion PROOF:

\underline{Corollary} If a series ∑ a_{n} converges, then lim a_{n} = 0 PROOF:

\underline{Conjecture} If lim a_{n} = 0, then ∑ a_{n} converges. TRUE or FALSE? PROOF

\underline{Theorem} lim a_{n} ≠ 0, then ∑ a_{n} diverges. PROOF:

Let’s prove that the COMPARISON TEST works for convergence. If ∑ a_{n} (where a_{n} > 0) converges, and |b_{n}| ≤ a_{n} ∀n, then ∑ b_{n} ALSO converges

\underline{Theorem} Let’s prove that the ROOT TEST works for convergence. Let ∑ a_{n} be a series and then if α = lim sup |a_{n}|^{1/n} and α < 1, then ∑ a_{n} converges.

\underline{Theorem} Let’s show how the RATIO TEST proves convergence.

\underline{Theorem}

15.1 ∑_{k=1}^{∞} \frac{1}{k^{p}} converges if and only if p > 1

Let’s use the Integral Test to prove this.

Homework Assignment (Due Friday November 1) §14, page 75: 14.1, 14.3, 14.4(b), 14.8, 14.13 §15, page 79: 15.1, 15.3, 15.6