New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here


by: Cassidy Grimes
Cassidy Grimes

GPA 3.51


Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

Class Notes
25 ?




Popular in Course

Popular in Mathematics (M)

This 3 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 554 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/229547/math-554-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.


Reviews for ANALYSIS I


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/26/15
MATH 554703 1 FALL 08 Lecture Note Set 4 Sequences and Series Example The following two results follow from the Principle of Induction and will useful in our study of convergence of sequences and series of real numbers n1 1 Zr71 r ifry l j0 r 2 1 no 3 1 a if a gt 0 amp n E W Bernoulli s inequality Defn Consider a sequence of points on from a metric space X d The following de nitions are used throughout the course 1 A sequence pn in a metric space X7d is convergent to p denoted by lam pn p means each e nbhd of p contains all but a nite number of terms bf the sequence We also use the shorter notation pn gt p when there is no ambiguity on the indices or the metric d pn is bounded means there is some element p E X and some real number M for this sequence so that dppn S M for all n E W pn is called Cauchy if for each 6 gt 0 there is an N 6 EV so that dpmpn lt 6 whenever m n 2 N 3 00 Example The following are examples of sequences of real numbers 1 121314 2 17r7r27r37 3 11r1rr21rr2r3m Lemma lim pn p if and only if for every 6 gt 07there exists N E Vso that if n Z W then dpnp lt e In short hand this reads Ve gt 0 EIN N E W 3 n 2 Ve gt dpnp lt 67 Proof Notice that if a statement is true except for at most a nite number of terms7 then there is a largest integer for which it is not true Take N to be that integer7s successor D Theorem If lim pn exists then it is unique TLHOO Proof Suppose that lim pn P1 and lim pn P2 and that P1 31 P2 Set 6 dP17P2 Now Hoe Hoe E gt 0 so there exists N17 such that if n 2 N1 then don7 P1 lt 6 Since the sequence converges to P27 we also have that there exists N27 such that if n 2 N2 then dme2 lt 6 Let N N1 N27 then N is larger than both N1 and N2 and so 510317132 3 dP17PN dltPN7 P2 lt 26 610317132 which gives a contradiction D Theorem Each convergent sequence is bounded Proof Suppose that lim pn p Let E 17 then there is an integer N such that pn E N5p if n 2 N Set M mlgxoi17dp17p7dp27p 7doN17p7 then the sequence is contained in the neighborhood NMp D Note i In the real numbers a set S is bounded if and only if there exists M gt 0 so that a S M for all a E S ii Not every bounded sequence is convergent For example the sequence an 1 is bounded but it is not convergent take 6 1 Theorem Each convergent sequence is Cauchy Special Properties of Sequences and Series for R Examples The following are important special cases of convergent sequences and series in the metric space R with the standard metric 1 1 lim naoo 71 Proof Use the Archimedean Principle 3n2 1 1m new 112 n 25 Hint Directly for a given 6 gt 07 use N max7674N1 where N1 is the cutoff7 for Example 17 ie any integer larger than 16 lfrlt 1 then r gt0 Proof If r 07 then the conclusion follows straight away Suppose that 0 lt lrl lt 17 then if b 1lrl71 we see that b gt 0 and lrl 11 b By Bernoulli7s inequality7 lrnl l 1 b 2 1 nb lnverting this inequality gives lr 7 0 S 11 71 By example 17 pick N so that 171 lt be ifn 2 N Hence7 10 OJ 1 n70lt7lt7lt if gtND lr l71nb nb 671717 4 lim 5n11 r ifsn1rr2r and rlt1 TL Note 5n Z rj 7 the sequence of partial sums of the geometric series j0 Proof If r 07 the conclusion follows immediately We may suppose then that 0 lt lrl lt 1 In this case7 we use the identity above7 ie 1 7 rquot1 1 7 r to see that 5 7 s 7rn117 r where s 117r Now7 given 6 gt 07 by example 3 there is an No such that n 2 N0 implies Tn lt1il7l D W 6 Combined with the displayed equation7 this gives lsn 7 sl lt E if n 2 N0 Theorem Properties of Limits Suppose that lim an a and lim bn b then 1 2 3 The limanbnab lim anbn ab If b y 0 th 139 an en 1m 7 n7gtoo bn b orem Suppose that lim an a then prove that lim an a Defn A sequence an is called monotone increasing if am 3 an Whenever m S n A sequence an is called monotone decreasing if an 3 am Whenever m S n The The orem Monotone sequences which are also bounded converge orem Suppose that lim an a and lim bn a If an 3 en 3 bn for all n E W then lim cn exists and equals a TLHOO The orem In R each Cauchy sequence is convergent General metric spaces which have this property are called complete metric spaces


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Allison Fischer University of Alabama

"I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

Steve Martinelli UC Los Angeles

"There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.