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                                                           Elementary	analysis	the	theory	of	calculus	2nd	edition	pdf
  Math	352Theory	of	CalculusSpring	2020	Course:	Math	352,	The	Theory	of	Calculus	Time:	MWF	1:00	pm	-	1:50	pm	Classroom:	Neckers	156	Textbook:	Elementary	Analysis:	The	Theory	of	Calculus,	2nd	edition,	by	Kenneth	A.	Ross,	Springer-Verlag.	There	is	a	free	PDF	version	of	the	textbook	here.	I	think	it	is	better	to	order	a	paper	copy,	but	it	is	up	to
  you.	Suggested	Supplement:	Your	old	calculus	textbook.	Instructor:	Professor	Sullivan	Office:	Neckers	385	Website:	E-mail:	Prof.Michael.Sullivan	(at)	gmail	(dot)	com	Phone:	453-6592	Office	hours:	MWF	10-12.	General	University	Info	You	are	responsible	for	all	the	material	in	Chapters	1-6	in	Elementary	Analysis:	The	Theory	of	Calculus,	2nd,	by
  Kenneth	A.	Ross,	Springer-Verlag	except	for	sections	6,	13,	21,	22,	and	35.	There	will	be	three	regular	exams	and	an	optional	final	exam.	Each	of	the	regular	exams	will	be	in	the	evening	and	you	will	have	two	hours.	Exam	1	will	cover	Chapters	1	&	2,	Exam	2	will	cover	Chapters	3	&	4,	and	Exam	3	will	cover	chapters	5	&	6.	No	books,	notes	or
  electronic	devices	are	premitted.	Homework	will	be	collected	weekly.	You	will	be	asked	to	give	short	presentations	to	the	class.	Each	of	the	three	regular	exams	will	count	as	25%	of	your	course	grade.	Homework	will	count	as	the	remaining	25%.	If	you	take	the	optional	final	it	will	count	as	20%	of	your	grade	with	the	other	exams	and	homework
  counting	as	20%	each.	Solutions	to	many	textbook	problems	can	be	found	on	the	internet.	I	know	how	to	find	them.	If	you	turn	in	work	that	is	not	your	own	I	will	lower	your	course	grade	by	one	letter	grade.	Cheating	on	an	exam	will	result	in	a	grade	of	F	in	the	course.	Homework	Assignments	Hwk	Set	1	Hwk	Set	2	Hwk	Set	3	Hwk	Set	4	Hwk	Set	5
  Hwk	Set	6	Hwk	Set	7	(Due	Monday	3/2.)	Hwk	Set	8	(Due	Friday	3/6.)	Hwk	Set	9	(Due	Monday	3/23.)	Hwk	Set	10	(Due	Friday	4/3.)	Hwk	Set	11	(Due	Wednesday	4/8.)	Hwk	Set	12	(Due	Monday	4/13.)	Hwk	Set	13	(Due	Monday	4/20.)	Hwk	Set	14	(Due	Firday	5/1.)	Handouts	Finite	and	Infinite	Sets	Algebraic	and	Transcendental	Numbers	Summary	of
  Calculus	Ordered	Fields	Triangle	Inequality	Completeness	Axiom	Sequences	and	Finite	Limits	Sequences	and	Infinite	Limits	Experimental	Field	Axioms	Worksheet	Cauchy	Sequences	(Section	10)	Subsequences	(Section	11)	Series	(Sections	14	&	15)	Review	Sheet	for	Test	1	Review	Sheet	for	Test	2	Review	Sheet	for	Test	3	Chapter	3	Continuous
  Functions	(Section	17)	Extreme	Theorems	(Section	18)	Uniform	Continuity	(Section	19)	Limits	of	Functions	(Section	20)	[In	Progress]	*Limit	Definitions	(Section	20)	*Limit	Rules	(Section	20)	Chapter	4	Power	Series	(Section	23)	Uniform	Convergence	(Section	24)	*Plots	for	24.3-5	*Example	from	class	Weierstrass	M-Test	(Section	25)	Begin	Online
  Here	Monday,	March	23:	Section	26	(part	1):	notes	video	webm	Suppliment:	Youtube	video	on	arctan	series	Suppliment:	Video	with	arctan	plot	(short)	in	webm	format	Wednesday,	March	25,	(Proof	of	Abel's	Theorem):	Section	26	(part	2):	notes	video:	mp4	webm	Friday,	March	27.	Section	28,	Part	1:	notes	video:	mp4	webm	Section	28,	Part	2:	notes
  video:	mp4	webm	Monday,	March	30.	Test	2.	7	pm	-	10	pm.	Wednesday,	April	1.	Section	29:	The	Mean	Value	Theorem.	Part	I.	notes	video:	mp4	webm	Section	29:	The	Mean	Value	Theorem.	Part	II.	notes	1	notes	2	webm	Suppliment	on	Concavity:	notes	webm	Friday,	April	3.	Section	30:	L'Hospital's	Rule.	notes	webm	1	webm	2	Optional	Reading:	On
  L'Hôpital's	Rule	by	Lorenzo	Sudan	Examples:	here	Monday,	April	6.	Section	31:	Taylor's	Theorem.	notes	webm	Wednesday,	April	8.	Section	31:	Binomial	Theorem.	notes	webm	Note:	You	can	skip	the	subsections	on	Newton's	Method	and	the	Secant	Method.	Friday,	April	10.	Chapter	6:	Integration.	Section	32.	notes	webm	drawing	webm	Monday,	April
  13.	Section	32.	notes	webm	Wednesday,	April	15.	Section	32.	notes	webm	notes	webm	Friday,	April	17.	Section	33.	notes	webm	Monday,	April	20.	Section	34.	notes	webm	footnote	Wednesday,	April	22.	Section	34.	notes	webm	Suppliment:	Substitution	Suppliment:	Integration	by	Parts	Optional	reading:	Trapiziodal	Rule	Error	Estimate	Formula	Friday,
  April	24.	Section	36:	notes	webm	Monday,	April	27.	Test	3????	Wednesday,	April	29.	Friday,	May	1.	Final	Exam:	To	be	arranged.	Books	You	Should	Read	Some	Day	The	History	of	the	Calculus	and	Its	Conceptual	Development	by	Carl	Boyer,	1949.	Infinitesimal:	How	a	Dangerous	Mathematical	Theory	Shaped	the	Modern	World	by	Amir	Alexander,
  2014.	Proofs	and	Refutations:	The	Logic	of	Mathematical	Discovery	by	Imre	Lakatos,	1976.	The	Analyst:	A	Discourse	Addressed	to	an	Infidel	Mathematician	by	George	Berkeley,	1734.	Want	more?	Advanced	embedding	details,	examples,	and	help!	Designed	for	students	having	no	previous	experience	with	rigorous	proofs,	this	text	on	analysis	can	be
  used	immediately	following	standard	calculus	courses.	It	is	highly	recommended	for	anyone	planning	to	study	advanced	analysis,	e.g.,	complex	variables,	differential	equations,	Fourier	analysis,	numerical	analysis,	several	variable	calculus,	and	statistics.	It	is	also	recommended	for	future	secondary	school	teachers.	A	limited	number	of	concepts
  involving	the	real	line	and	functions	on	the	real	line	are	studied.	Many	abstract	ideas,	such	as	metric	spaces	and	ordered	systems,	are	avoided.	The	least	upper	bound	property	is	taken	as	an	axiom	and	the	order	properties	of	the	real	line	are	exploited	throughout.	A	thorough	treatment	of	sequences	of	numbers	is	used	as	a	basis	for	studying	standard
  calculus	topics.	Optional	sections	invite	students	to	study	such	topics	as	metric	spaces	and	Riemann-Stieltjes	integrals.	For	over	three	decades,	this	best-selling	classic	has	been	used	by	thousands	of	students	in	the	United	States	and	abroad	as	a	must-have	textbook	for	a	transitional	course	from	calculus	to	analysis.	It	has	proven	to	be	very	useful	for
  mathematics	majors	who	have	no	previous	experience	with	rigorous	proofs.	Its	friendly	style	unlocks	the	mystery	of	writing	proofs,	while	carefully	examining	the	theoretical	basis	for	calculus.	Proofs	are	given	in	full,	and	the	large	number	of	well-chosen	examples	and	exercises	range	from	routine	to	challenging.The	second	edition	preserves	the	book’s
  clear	and	concise	style,	illuminating	discussions,	and	simple,	well-motivated	proofs.	New	topics	include	material	on	the	irrationality	of	pi,	the	Baire	category	theorem,	Newton's	method	and	the	secant	method,	and	continuous	nowhere-differentiable	functions.Review	from	the	first	edition:"This	book	is	intended	for	the	student	who	has	a	good,	but	naïve,
  understanding	of	elementary	calculus	and	now	wishes	to	gain	a	thorough	understanding	of	a	few	basic	concepts	in	analysis....	The	author	has	tried	to	write	in	an	informal	but	precise	style,	stressing	motivation	and	methods	of	proof,	and	...	has	succeeded	admirably."—MATHEMATICAL	REVIEWS	Page	2	Undergraduate	Texts	in	Mathematics	Page	3
  Undergraduate	Texts	in	Mathematics	Series	Editors:	Sheldon	Axler	San	Francisco	State	University,	San	Francisco,	CA,	USA	Kenneth	Ribet	University	of	California,	Berkeley,	CA,	USA	Advisory	Board:	Colin	C.	Adams,	Williams	College,	Williamstown,	MA,	USA	Alejandro	Adem,	University	of	British	Columbia,	Vancouver,	BC,	Canada	Ruth	Charney,
  Brandeis	University,	Waltham,	MA,	USA	Irene	M.	Gamba,	The	University	of	Texas	at	Austin,	Austin,	TX,	USA	Roger	E.	Howe,	Yale	University,	New	Haven,	CT,	USA	David	Jerison,	Massachusetts	Institute	of	Technology,	Cambridge,	MA,	USA	Jeffrey	C.	Lagarias,	University	of	Michigan,	Ann	Arbor,	MI,	USA	Jill	Pipher,	Brown	University,	Providence,	RI,
  USA	Fadil	Santosa,	University	of	Minnesota,	Minneapolis,	MN,	USA	Amie	Wilkinson,	University	of	Chicago,	Chicago,	IL,	USA	Undergraduate	Texts	in	Mathematics	are	generally	aimed	at	third-	and	fourth-	year	undergraduate	mathematics	students	at	North	American	universities.	These	texts	strive	to	provide	students	and	teachers	with	new
  perspectives	and	novel	approaches.	The	books	include	motivation	that	guides	the	reader	to	an	appreciation	of	interrela-	tions	among	different	aspects	of	the	subject.	They	feature	examples	that	illustrate	key	concepts	as	well	as	exercises	that	strengthen	understanding.	For	further	volumes:	Page	4	Kenneth	A.	Ross	Elementary	Analysis	The	Theory	of
  Calculus	Second	Edition	In	collaboration	with	Jorge	M.	Lo´pez,	University	of	Puerto	Rico,	R´ıo	Piedras	123	Page	5	Kenneth	A.	Ross	Department	of	Mathematics	University	of	Oregon	Eugene,	OR,	USA	ISSN	0172-6056	ISBN	978-1-4614-6270-5	ISBN	978-1-4614-6271-2	(eBook)	DOI	10.1007/978-1-4614-6271-2	Springer	New	York	Heidelberg	Dordrecht
  London	Library	of	Congress	Control	Number:	2013950414	Mathematics	Subject	Classification:	26-01,	00-01,	26A06,	26A24,	26A27,	26A42	©	Springer	Science+Business	Media	New	York	2013	This	work	is	subject	to	copyright.	All	rights	are	reserved	by	the	Publisher,	whether	the	whole	or	part	of	the	material	is	concerned,	specifically	the	rights	of
  translation,	reprinting,	reuse	of	illustrations,	recitation,	broadcasting,	reproduction	on	microfilms	or	in	any	other	physical	way,	and	transmission	or	information	storage	and	retrieval,	electronic	adaptation,	computer	software,	or	by	similar	or	dissim-	ilar	methodology	now	known	or	hereafter	developed.	Exempted	from	this	legal	reservation	are	brief
  excerpts	in	connection	with	reviews	or	scholarly	analysis	or	material	supplied	specifically	for	the	pur-	pose	of	being	entered	and	executed	on	a	computer	system,	for	exclusive	use	by	the	purchaser	of	the	work.	Duplication	of	this	publication	or	parts	thereof	is	permitted	only	under	the	provisions	of	the	Copyright	Law	of	the	Publisher’s	location,	in	its
  current	version,	and	permission	for	use	must	always	be	obtained	from	Springer.	Permissions	for	use	may	be	obtained	through	RightsLink	at	the	Copyright	Clearance	Center.	Violations	are	liable	to	prosecution	under	the	respective	Copyright	Law.	The	use	of	general	descriptive	names,	registered	names,	trademarks,	service	marks,	etc.	in	this	publi-
  cation	does	not	imply,	even	in	the	absence	of	a	specific	statement,	that	such	names	are	exempt	from	the	relevant	protective	laws	and	regulations	and	therefore	free	for	general	use.	While	the	advice	and	information	in	this	book	are	believed	to	be	true	and	accurate	at	the	date	of	publication,	neither	the	authors	nor	the	editors	nor	the	publisher	can
  accept	any	legal	responsibility	for	any	errors	or	omissions	that	may	be	made.	The	publisher	makes	no	warranty,	express	or	implied,	with	respect	to	the	material	contained	herein.	Printed	on	acid-free	paper	Springer	is	part	of	Springer	Science+Business	Media	(www.springer.com)	Page	6	Preface	Preface	to	the	First	Edition	A	study	of	this	book,	and
  espe-	cially	the	exercises,	should	give	the	reader	a	thorough	understanding	of	a	few	basic	concepts	in	analysis	such	as	continuity,	convergence	of	sequences	and	series	of	numbers,	and	convergence	of	sequences	and	series	of	functions.	An	ability	to	read	and	write	proofs	will	be	stressed.	A	precise	knowledge	of	definitions	is	essential.	The	be-	ginner
  should	memorize	them;	such	memorization	will	help	lead	to	understanding.	Chapter	1	sets	the	scene	and,	except	for	the	completeness	axiom,	should	be	more	or	less	familiar.	Accordingly,	readers	and	instructors	are	urged	to	move	quickly	through	this	chapter	and	refer	back	to	it	when	necessary.	The	most	critical	sections	in	the	book	are	§§7–12	in
  Chap.	2.	If	these	sections	are	thoroughly	digested	and	understood,	the	remainder	of	the	book	should	be	smooth	sailing.	The	first	four	chapters	form	a	unit	for	a	short	course	on	analysis.	I	cover	these	four	chapters	(except	for	the	enrichment	sections	and	§20)	in	about	38	class	periods;	this	includes	time	for	quizzes	and	examinations.	For	such	a	short
  course,	my	philosophy	is	that	the	students	are	relatively	comfortable	with	derivatives	and	integrals	but	do	not	really	understand	sequences	and	series,	much	less	sequences	and	series	of	functions,	so	Chaps.	1–4	focus	on	these	topics.	On	two	v	Page	7	Preface	vi	or	three	occasions,	I	draw	on	the	Fundamental	Theorem	of	Calculus	or	the	Mean	Value
  Theorem,	which	appears	later	in	the	book,	but	of	course	these	important	theorems	are	at	least	discussed	in	a	standard	calculus	class.	In	the	early	sections,	especially	in	Chap.	2,	the	proofs	are	very	detailed	with	careful	references	for	even	the	most	elementary	facts.	Most	sophisticated	readers	find	excessive	details	and	references	a	hindrance	(they
  break	the	flow	of	the	proof	and	tend	to	obscure	the	main	ideas)	and	would	prefer	to	check	the	items	mentally	as	they	proceed.	Accordingly,	in	later	chapters,	the	proofs	will	be	somewhat	less	detailed,	and	references	for	the	simplest	facts	will	often	be	omit-	ted.	This	should	help	prepare	the	reader	for	more	advanced	books	which	frequently	give	very
  brief	arguments.	Mastery	of	the	basic	concepts	in	this	book	should	make	the	analysis	in	such	areas	as	complex	variables,	differential	equations,	numerical	analysis,	and	statistics	more	meaningful.	The	book	can	also	serve	as	a	foundation	for	an	in-depth	study	of	real	analysis	given	in	books	such	as	[4,33,34,53,62,65]	listed	in	the	bibliography.	Readers
  planning	to	teach	calculus	will	also	benefit	from	a	careful	study	of	analysis.	Even	after	studying	this	book	(or	writing	it),	it	will	not	be	easy	to	handle	questions	such	as	“What	is	a	number?”	but	at	least	this	book	should	help	give	a	clearer	picture	of	the	subtleties	to	which	such	questions	lead.	The	enrichment	sections	contain	discussions	of	some	topics
  that	I	think	are	important	or	interesting.	Sometimes	the	topic	is	dealt	with	lightly,	and	suggestions	for	further	reading	are	given.	Though	these	sections	are	not	particularly	designed	for	classroom	use,	I	hope	that	some	readers	will	use	them	to	broaden	their	horizons	and	see	how	this	material	fits	in	the	general	scheme	of	things.	I	have	benefitted	from
  numerous	helpful	suggestions	from	my	col-	leagues	Robert	Freeman,	William	Kantor,	Richard	Koch,	and	John	Leahy	and	from	Timothy	Hall,	Gimli	Khazad,	and	Jorge	L´opez.	I	have	also	had	helpful	conversations	with	my	wife	Lynn	concerning	grammar	and	taste.	Of	course,	remaining	errors	in	grammar	and	mathematics	are	the	responsibility	of	the
  author.	Several	users	have	supplied	me	with	corrections	and	suggestions	that	I’ve	incorporated	in	subsequent	printings.	I	thank	them	all,	Page	8	Preface	vii	including	Robert	Messer	of	Albion	College,	who	caught	a	subtle	error	in	the	proof	of	Theorem	12.1.	Preface	to	the	Second	Edition	After	32	years,	it	seemed	time	to	revise	this	book.	Since	the	first
  edition	was	so	successful,	I	have	retained	the	format	and	material	from	the	first	edition.	The	num-	bering	of	theorems,	examples,	and	exercises	in	each	section	will	be	the	same,	and	new	material	will	be	added	to	some	of	the	sections.	Every	rule	has	an	exception,	and	this	rule	is	no	exception.	In	§11,	a	theorem	(Theorem	11.2)	has	been	added,	which
  allows	the	sim-	plification	of	four	almost-identical	proofs	in	the	section:	Examples	3	and	4,	Theorem	11.7	(formerly	Corollary	11.4),	and	Theorem	11.8	(formerly	Theorem	11.7).	Where	appropriate,	the	presentation	has	been	improved.	See	es-	pecially	the	proof	of	the	Chain	Rule	28.4,	the	shorter	proof	of	Abel’s	Theorem	26.6,	and	the	shorter	treatment
  of	decimal	expansions	in	§16.	Also,	a	few	examples	have	been	added,	a	few	exercises	have	been	modified	or	added,	and	a	couple	of	exercises	have	been	deleted.	Here	are	the	main	additions	to	this	revision.	The	proof	of	the	irrationality	of	e	in	§16	is	now	accompanied	by	an	elegant	proof	that	π	is	also	irrational.	Even	though	this	is	an	“enrichment”
  section,	it	is	especially	recommended	for	those	who	teach	or	will	teach	pre-	college	mathematics.	The	Baire	Category	Theorem	and	interesting	consequences	have	been	added	to	the	enrichment	§21.	Section	31,	on	Taylor’s	Theorem,	has	been	overhauled.	It	now	includes	a	discussion	of	Newton’s	method	for	approximating	zeros	of	functions,	as	well	as
  its	cousin,	the	secant	method.	Proofs	are	provided	for	theorems	that	guarantee	when	these	approximation	methods	work.	Section	35	on	Riemann-Stieltjes	integrals	has	been	improved	and	expanded.	A	new	section,	§38,	contains	an	example	of	a	continuous	nowhere-	differentiable	function	and	a	theorem	that	shows	“most”	continuous	functions	are
  nowhere	differentiable.	Also,	each	of	§§22,	32,	and	33	has	been	modestly	enhanced.	It	is	a	pleasure	to	thank	many	people	who	have	helped	over	the	years	since	the	first	edition	appeared	in	1980.	This	includes	David	M.	Bloom,	Robert	B.	Burckel,	Kai	Lai	Chung,	Mark	Dalthorp	(grandson),	M.	K.	Das	(India),	Richard	Dowds,	Ray	Hoobler,	Page	9	Preface
  viii	Richard	M.	Koch,	Lisa	J.	Madsen,	Pablo	V.	Negr´on	Marrero	(Puerto	Rico),	Rajiv	Monsurate	(India),	Theodore	W.	Palmer,	Ju¨rg	Ra¨tz	(Switzerland),	Peter	Renz,	Karl	Stromberg,	and	Jesu´s	Sueiras	(Puerto	Rico).	Special	thanks	go	to	my	collaborator,	Jorge	M.	Lo´pez,	who	pro-	vided	a	huge	amount	of	help	and	support	with	the	revision.	Working	with
  him	was	also	a	lot	of	fun.	My	plan	to	revise	the	book	was	sup-	ported	from	the	beginning	by	my	wife,	Ruth	Madsen	Ross.	Finally,	I	thank	my	editor	at	Springer,	Kaitlin	Leach,	who	was	attentive	to	my	needs	whenever	they	arose.	Especially	for	the	Student:	Don’t	be	dismayed	if	you	run	into	material	that	doesn’t	make	sense,	for	whatever	reason.	It
  happens	to	all	of	us.	Just	tentatively	accept	the	result	as	true,	set	it	aside	as	something	to	return	to,	and	forge	ahead.	Also,	don’t	forget	to	use	the	Index	or	Symbols	Index	if	some	terminology	or	notation	is	puzzling.	Page	10	Contents	Preface	v	1	Introduction	1	1	The	Set	N	of	Natural	Numbers	.	.	.	.	.	.	.	.	.	.	.	.	1	2	The	Set	Q	of	Rational	Numbers	.	.	.	.	.	.	.	.
  .	.	.	6	3	The	Set	R	of	Real	Numbers	.	.	.	.	.	.	.	.	.	.	.	.	.	13	4	The	Completeness	Axiom	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	20	5	The	Symbols	+∞	and	−∞	.	.	.	.	.	.	.	.	.	.	.	.	.	.	28	6	*	A	Development	of	R	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	30	2	Sequences	33	7	Limits	of	Sequences	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	33	8	A	Discussion	about	Proofs	.	.	.	.	.	.	.	.	.	.	.	.	.	.	39	9	Limit	Theorems	for	Sequences	.	.	.	.	.
  .	.	.	.	.	.	.	45	10	Monotone	Sequences	and	Cauchy	Sequences	.	.	.	.	56	11	Subsequences	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	66	12	lim	sup’s	and	lim	inf’s	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	78	13	*	Some	Topological	Concepts	in	Metric	Spaces	.	.	.	83	14	Series	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	.	95	15	Alternating	Series	and	Integral	Tests	.	.	.	.	.	.	.	.	105	16	*	Decimal	Expansions	of
  Real	Numbers	.	.	.	.	.	.	.	109	ix
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