section9notes

section9notes - Lecture 9 • Definition of a bounded...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 9 • Definition of a bounded sequence. • Theorem: If (sn ) converges, it is bounded. • Give a counterexample to the statement: If (sn ) is bounded, then it converges. • Theorem: If sn → s and tn → t, then sn + tn → s + t. • Theorem: If sn → s and tn → t, then sn tn → st. • Theorem: If sn → s and sn = 0 ∀n and s = 0, then 1 sn → 1. s • Corollary: If sn → s and tn → t, sn = 0 ∀n, and s = 0, then tn sn t → s. • Example: Use the above theorems to find n3 + 6 n2 + 7 . n→∞ 4n3 + 3n − 4 lim • Make sure to read 9.7(a)-(d), and be able to reproduce the proofs. • Definition of a sequence diverging to ±∞. What is the difference between this and merely being unbounded? • Prove n2 − 3n diverges to ∞. • Prove n2 +3 n+1 diverges to ∞. • Theorem: If (sn ) diverges to ∞ and lim tn exists and is positive, then (sn tn ) diverges to ∞. • Theorem: (sn ) diverges to ∞ iff 1 sn → 0. • Theorem: If ∃M so that ∀n ≥ M sn ≤ tn , and if sn → s and tn → t, then s ≤ t. 1 ...
View Full Document

Ask a homework question - tutors are online