L15 Monotonic Sequence Theorem

L15 Monotonic Sequence Theorem - and is bounded below then...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 15: Sequences (II) Monotonic Sequence Thmorem A sequence is increasing if a n < a n +1 ; decreasing if a n > a n +1 : A sequence is mono- tonic if it is either increasing or decreasing. A sequence is bounded above if there is a number M such that a n ± M for all n ² 1; bounded below if there is a number m such that m ± a n for all n ² 1. A sequence is bounded if it’s bounded above and below. Monotonic Sequence Thm: Every bounded, monotonic sequence is convergent. Alternative statements of the theorem: If a sequence f a n g is monotonically increasing and is bounded above, then f a n g is convergent. If a sequence f a n g is monotonically decreasing
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: and is bounded below, then f a n g is convergent. 1 ex. f a n g is a monotonic increasing sequence such that a 1 = 2 ;a n +1 = 1 2 ( a n + 6) for n ± 2 : and a n < 6 for all n . Find the limit of the sequence. ex. f a n g is a monotonically decreasing sequence such that a 1 = 2 ;a n +1 = 1 3 ² a n for n ± 1 : and < a n ³ 2 for all n . Find the limit of the sequence. 2 NYTI: Sequence p 2 ; p 2 p 2 ; q 2 p 2 p 2 ::: is mono-tonically increasing and bounded above. Find the limit of the sequence. 3...
View Full Document

This note was uploaded on 02/14/2012 for the course MAC 2312 taught by Professor Bonner during the Spring '08 term at University of Florida.

Page1 / 3

L15 Monotonic Sequence Theorem - and is bounded below then...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online