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# ADVANCED CALCULUS MTH 311

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This 57 page Class Notes was uploaded by Miss Johan Jacobson on Monday October 19, 2015. The Class Notes belongs to MTH 311 at Oregon State University taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/224455/mth-311-oregon-state-university in Mathematics (M) at Oregon State University.

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Infinite Sequences of Real Numbers AKA ordered lists DEF An infinite sequence of real numbers is a functionf N gt R Usually infinite sequences are written as lists such as x71117 xn 951352353 where x fn DEF An infinite sequence xn01 has the real number a as a limit if given an real number s gt 0 there is a corresponding N e N such that ngtNgt xn a lt8 in which case we write yran a orxn gt 6161811 gt oo DEF If a sequence has a nite limit we say it converges Facts The sequence ln converges and has limit 0 A sequence may not have a limit Limits are unique if they exist Explicit Calculations With Limits The process of taking a limit interacts as you would expect with the basic operations x and and with the order properties of real numbers Th Suppose xn and yn are convergent sequences in R and c e R Then the limits on the left all exist and have the indicated value mm yngt my imam i xnyn 1193quot 39lilgloyn If in addition limnsooyn 2 0 then lug Z t Th Suppose xn and yn are convergent sequences in R and xn lt y for all large n or xn 5 y for all large n Then limxn S limyn n gtOO n gtOO The Squeeze Law AKA The Three Sequences Theorem Th Suppose xn yn and 2 are sequences in R with xn 5 y 5 2 for all large n Bligh and 1332 and Then Egg and all three limits are equal Corollaries 2 If xn is bounded which means 3M gt 0 such that xn 5 Mfor all n and ifyn gt O as n gt 00 then Elimnsooxnyn O The Dilemma With Limits In significant applications in which limits occur you generate a sequence xn whose terms get quotcloser and closer to the solution of a difficult practical or theoretical problem that you cannot solve by more elementary means Your hope is that quotcloser and closer means the sequence converges and that its limit say a is the solution to your problem Here is the dilemma You don t know a If you did you wouldn t need the sequence If you don t know a then you can t use the definition of a limit to check that the sequence has limit a What are you to do You need ways to guarantee that a sequence converges without knowing its limit in advance Stay turned Basic Properties a Sequence May Have DEF A sequence xn of real numbers is bounded above if M e R such that x 5 Mfor all n bounded below if Elm e R such that xn 2 m for all 11 bounded if it is both bounded above and bounded below Facts 1 xn is bounded ltgt 3M 6 R such that xn 5 M for all n 2 Convergent sequences are bounded 3 Bounded sequences need not converge 4 Bounded sequences often contain convergent subsequences 00 DEF A subsequence of a sequence xn1 is a sequence of the form xnk2o1 xnlaxn27 xn37quot39 wherel 711 ltn2ltn3 lt Increasing and Decreasing Sequences DEF A sequence xn of real numbers is increasing ifxl 5 x2 5 x3 55 xn 5 decreasing ifxl 2 x2 2x3 2 2x 2 strictly increasing ifxl lt x2 lt x3 lt lt xn lt strictly decreasing if x1gtX2gtX3 gt gtxngt monotone if it is either increasing or decreasing Monotone sequences are important because they give one way out of our dilemma Th Monotone Convergence Theorem Every bounded monotone sequence of real numbers converges Th MCT full disclosure version 1 An increasing sequence that is bounded above converges to is least upper bound 2 A decreasing sequence that is bounded below converges to its greatest lower bound 3 An increasing sequence that is not bounded above diverges to 00 4 A decreasing sequence that is not bounded below diverges to oo Very Important Consequences of Monotonicity Cantor s Nested Intervals Theorem If Inn N is a sequence of nonempty closed bounded nested intervals nested means 11 3 12 3 13 3 m then mneN In 5 Q Furthermore if In length ofIn gt O as n gt 00 then may 1 is a single real number Th Every sequence contains a monotone subsequence The BolzanoWeierstrass Theorem Every bounded sequence of real numbers has contains a convergent subsequence Cauchy Criterion for Convergence DEF A sequence xn01 is a Cauchy sequence if given any 8 gt O 3 Nsuch that n m gtNgt xn xm lt 8 Facts 1 Convergent sequences are Cauchy 2 Cauchy sequences are bounded 3 Theorem with a capital T A sequence of real numbers converges if and only if it is a Cauchy sequence This theorem provides another way out of the limit dilemma Limits of Functions Part I 7 twoesided limits STANDING CONVENTION f domf gt R and domf D I which is a nonempty open interval that contains the point a DEF The function f has limit L E R as at approaches a if given any a gt 0 there exists a 6 gt 0 such that 0ltlxiallt6 gt lfac7Lllta Notation If f has limit L as at approaches a we write limxgaf L or gtLasxgta DEF If a function has a nite limit we say it converges to its limit Facts lingLa mac b ma b A function may not have a limit as as A 1 Limits are unique if they exist What Happens AT ac a does NOT affect the limit as as A a Only the behavior of the function near a but not at 1 determines whether there is a lim1t at a The following result is used7 often implicitly7 in many limit calculations Th lf domf D I a domg a 91 forallacEIa then 3 lim f lim 9 xgta xgta Sequential Characterization of Limits of Functions The following result enables us to transfer most everything we know about limits of sequences to corresponding results about limits of functions Th SCLF The following are equivalent 1 Hlimxna ac L 2 For every sequence 1n1 in I a with limit a EIlimnaoof M 7 Remarks 1 This theorem also is true for the in nite limits L too andor for limits as at approaches too Of course we haven t de ned these limits yet but you should be able to 2 In the theorem when a E R the interval I can be any open interval con taining a In the case that as A 00 I can be any open interval of the formb What about when as A 700 The Algebra of Functions Given functions f X A R and g X gt R and a E R we de ne their sum difference scalar multiple product and quotient which are denoted ref f f97 fig 04f f9 9 sp ectively by as follows 1 f 9 is the function whose value at at is f 9 90 Hf 900 2 f 7 g is the function whose value at at is f 00 Hf 7990 3 af is the function whose value at at is af 1 af 1 4 fg is the function Whose value at at is f9 1 f 19 90 5 fg is the function Whose value at at is f f 1 7 as g 9 x The domain of each of the functions f g f 7 g af fg is dom dom g and the domain of fg is ac E dom dom g g 7E 0 Explicit Calculations With Limits The process of taking a limit interacts as you would expect With the basic operations 7 gtlt and and With the order properties of real numbers Th Suppose Hlimx ha f and Hlimx nag and c E R Then the limits on the left all exist and have the indicated value ggltxltxgtgltxgtgt mnmmgm memo we mama munimage If in addition limx mg g 7E 0 then hm new inhuman xgta g liming g Th Suppose f and g have limits as as A a and lt 91 for all at E I a or S 91 for all at E I a Then limx ha f S limxnag The Squeeze Law AKA The Three Functions Theorem Th Suppose f 91 and h are realevalued functions With f S g S h for all large at E I a 3 lim f and 3 lim h and lim f lim h xgta xgta xgta xgta Then 3 lim 9 and all three limits are equal xgta Corollaries 1 S hac A 0 gt Eliminag 0 2 If f is bounded near a Which means 3 M gt 0 such that lf S M for all at E I a and ifg A 0 as at A a then Hlimxna f acg 0 Oneesided Limits DEF righthand limit If dom contains some open interval With left ende point a E R and L E R then 7 133 f at i L means Given any a gt 0 3 6 gt 0 such that 0ltx7alt6gt fac7Llta DEF lefthand limit LTR Th Let I be an open interval7 a 6 I7 and dom D I a Then 3 lim f 3 lim f 7 3 lim f and lim f lim f xgta Iaai 1Ha Iaai 1Ha in Which case all three limits are equal Notation limf L f 12 and limi f L f 27 Remark All the algebraic limit laws and squeeze laws hold When thsided limits are replaced by oneesided limits Limits Involving ln nity DEF nite limit as as A 00 If dom contains an open interval of the form 07 00 for some 0 and L E R then 1320 x L means Given any a gt 0 3 M gt 0 such that acgtM gt fac7L lta DEF nite limit as as A 700 LTR Remark All the algebraic limit laws and squeeze laws hold for nite limits at too DEF limit 00 as at gt a 6 R If dom D I a then 13 f at 00 means Given any M E R 3 6 gt 0 such that 0ltlxiallt6 gt facgtM Remark WLOG you can assume in applying this de nition that M gt M0 for any convenient M0 Proof CDP It is left to you to give precise de nitions for the limits 22f m 7007 123 f m my Eglmf m too using the foregoing de nitions as models CDP Continuous Functions DEF Let f E gt R and a E E The function f is continuous at a if for every 5 gt 0 there exists a 6 gt 0 such that xEEand rial gt fac7fa lt5 If S C E and f is continuous at each point in 57 then we say f is continuous on S If f is continuous at each point in its domain we say f is continuous Facts 1 f b is continuous for any xed real number I 2 f at is continuous 3 f is continuous Continuity can be characterized sequentially Th Let f E gt R and a E E The following are equivalent 1 f is continuous at a 2 For each sequence in E with limit a the sequence f has limit f a Cor f is not continuous at a if 3 a sequence in E with limit a such that f does not have limit f a Fact The function f is continuous The theorem and the algebraic limit laws for sequences yield Th LetfEARandgEHRbecontinuousataEEandletaElR Then the functions f 97 f 7 97 af and fg7 are continuous at a Moreover7 if g 1 7E 07 the function fg is continuous at 1 Cor All polynomial functions and rational functions are continuous Very often continuity can be expressed conveniently in terms of limits Th Let f E gt R and a E E If 1 belongs to an open interval J and J C E7 then the following are equivalent 1 f is continuous at a 2 Hm f x f 11 Th Let I be an interval of any type with endpoints a lt f I gt R The following are equivalent 1 f is continuous on I 2 limx a f at f a at each point a E I with oneesided limits understood at the endpoints of I that belong to I Continuity of Composite Functions DEF If f X A Y and g Y gt Z are functions7 the function g o f X A Z de ned by 90 f 90 9fr for each at E X and read 9 composed with f is called a composite function Facts 1 Even ifgof and fog are both de ned it is usually the case that gof 7E fog Th Let E7 F7 and G be subsets or R Let f E A F7 9 F gt G and a E E Then 1 If f is continuous at a and g is continuous at b f a then 9 o f is continuous at a If f is continuous and g is continuous7 then 9 o f is continuous More About Sups and lnfs DEF Let E be a nonempty subset of real numbers 1 If E in not bounded above by de nition supE 00 2 If E in not bounded below by de nition infE 700 As a consequence of these conventions and the CA for R Every nonempty set E C R has a supremum and an in mum Facts 1 For E 7E 0 E is bounded abovelt 00 4 sup E 2 For E 7E 0 3 in E with son A sup E If supE E the sequence can be chosen strictly increasing 3 What is the sitution for inf E Properties a Function May Have DEF A function f E A R is bounded above if 3 M E R such that f S M for all at E E bounded below if 3 m E R such that f 2 m for all at E E bounded if it is bounded above and bounded below So f is bounded 4 3 M E R such that lf S M for all at E E Equivalentlya function is bounded above below if its range ran is bounded above below DEF A function f E A R is increasing if 11 12 E E and 11 lt 12 gt f 11 S f 352 decreasing if 11 12 E E and 1 gt 12 gt f 11 2 f 902 strictly increasing if 11 12 E E and 11 lt 12 f1 lt f2 strictly decreasing 1 12 E E and 1 gt 12 gt f1gt 902 monotone if it is either increasing or decreasing Fundamental Properties of Continuous Functions Th Sign Preserving Properly Let f E gt R be continuous at a E E If f a gt 07 then there is a 6 gt 0 such that xEEand lxiallt6 gt facgt0 Th Extreme or MaxiMih Value Theorem Let I be a closed7 bounded interval and f I A R be continuous Then f is bounded and f assumes its maximum and mininum values on I That is7 there are points a and in I such that fa fx f forallacEI In other words7 the range of f is a bounded set and contains its supremum and in mum Th A Intermediate Value Theorem Let I 1 b be a closed7 bounded interval and f I A R be continuous If y is any value strictly between f a and f b7 then there is a point at E 11 such that f y Here are two equivalent formulations of the lVT Th B Intermediate Value Theorem Let I 1 b be a closed7 bounded interval and f I gt R be continuous such that f a f b lt 0 Then there is a point at E 11 such that f 0 Th C Intermediate Value Theorem Let I be an interval of any type open7 closed7 hal open7 bounded or unbounded and f I A R be continuous Then the range of f is an interval Th nth roots exist Let n E N The function f 000 A 000 With f at is continuous7 strictly increasing and onto hence7 invertible Cor For each n E N and real number y 2 0 there is a unique nonnegative real number at such that at y Notation If y gt 07 the unique at above is the positive nth root of y denoted by y The Exponential Function on Q Let a gt 0 By de nition Step 1 a0 1 Step 2 For n E N 1 a aaa7 a 7 a nrfactors 111 Va Step 3 For q E Q write q mn with m E Z and n E N Then 1 7 a1 m Facts 1 The de nition in Step 3 makes sense That is7 if q mn m n with m m E Z and n n E N then 11 aquot 11 my 2 The usual rules of exponents hold aqq ag7 aiq 1aq am aqq abq aqbq b gt 0 3 The function f Q A R de ned by f q 1 7 is continuous and strictly increasing if a gt 1 and is strictly decreasing if 0 lt a lt Rational Power Functions 1 For each xed value of at gt 0 think of ac a and each q E Q the number acq has already been de ned It is just a matter of your point of view So we now know what the function f 000 A R given by f 1 7 means for any rational power q q mn With m E Z and n an odd number including n 17 then proon each at E 7007 00 has a unique nth root 11 and by de nition acq 11 n For such q7 f 1quot has domain 70000 3 The function f 1quot is continuous on its domain Uniform Continuity DEF Let E C R A function f E A R is uniformly continuous on E if for every 5 gt 0 there exists a 6 gt 0 such that 116Eandlxiac llt6 gt lf fllt5 Facts 1 f uniformly continuous on E implies f is continuous on E 2 Continuity and uniform continuity on a set are not equivalent concepts Th Let f E A R be uniformly continuous Then f maps Cauchy sequences in E onto Cauchy sequences in R The Exponential Function on R 1 Fix a gt 0 f q 1 7 is uniformly continuous on Og Q for any bounded interval m C R 2 If at E R and qn is any sequence in Q With limit at then 3 lim aqquot ngtoo 3 The limit in 2 depends only upon at and not upon the particular sequence qn With limit at The foregoing facts justify the following de nition DEF Fix a gt 0 Let at E R By de nition as lim aqquot ngtoo 10 Where qn is any sequence in Q With limit at In other words 11 is the unique continuous extension of 1 7 from Q to R Th Fix a gt 0 With 1 7E 1 The function f R A R de ned by 11 is positive continuous has range 0 00 satis es the usual rules of exponents and is strictly increasing When a gt 1 and strictly decreasing When 0 lt a lt 1 Uniform Continuity cont DEF Let E C R The Closure of E denoted E consists of all points at E R that are limits of sequences in E That is 3 a sequence in E With limit at DEF Let E C R E is Closed if E E Facts 1 E 0 1 has closure E 0 1 2 Q lR another version of Q is dense in R 3 E a 1 Q has closure E 11 Th A If E C R is closed and bounded and f E A R is continuous then f is uniformly continuous on E Th B Let E C R and f E A R be uniformly continuous on E Then f has a unique extension by continuity to E That is there is a unique continuous function g E A R Whichin fact is uniformly continuous on E such that 935 for all at E E Number Systems N 123 the positive integers Z 3 2 10123 the integers Q pq e Z with q 2 O the rational numbers R 2 numbers expressible by finite or unending decimal expansions makes sense in N and make sense in Z and and x and make sense in Q and R lt makes sense in N and Z and Q and R All the usual rules of arithmetic and inequalities hold in Q and R Some but not all hold in N and Z Postulate 1 Field Axioms in WRW gt the usual arithmetic rules Postulate 2 Order Axioms in WRW gt the usual rules for inequalities Absolute Value of a Real Number a if a 2 O lal a If a 5 O dist0a on the number line Basic Properties For all ab e R a 2 OWith iffa 0 la bl lb a a 9 5 a b The triangle inequality The triangle inequality implies llal W 5 la b a 9 2 distab on the number line Finally aSMltgt M a M What is a Mathematical Theory such as calculus or geometry A logical development of a subject that starts from basic assumptions often called axioms and postulates that are taken as selfevident Thereafter the only statements accepted as true or false are those that have been so proven reasoning from the axioms and using accepted rules of logical argumentation The logical consequences of the axioms are called Theorems Propositions Lemmas according to a somewhat arbitrary scheme The logical argumentation leading to the theorems etc is called quotproof39 Why is a Theory Needed Some mathematical statements that appear true on geometric grounds or on physical grounds or according to your intuition turn out to be false Some Plausible Statements That Are Not True The rational numbers fill up the number line i i L i The serIes 1 2 3 4 5 converges The function lnx has a horizontal asymptote Three Main Types of Proof 1 Mathematical Induction A basic property of the positive integers and far more powerful than first meets the eye 2 Direct Proof To prove If H then C you assume that H is true and use accepted principles of logical argumentation using the axioms and already established facts to deduce that C must logically follow 3 Indirect Proof AKA Proof by Contradiction To prove If H then C you start by assuming H and that C is FALSE Then you use accepted principles of logical argumentation to reach a contradiction This proves that H implies C A contradiction is a statement that is know to be false The statement thatx 5 and x gt 5 is a contradiction Why is indirect proof a valid argument Postulate 3 The WellOrdering Axiom Every nonempty subset of positive integers has a smallest element Theorem Principle of Mathematical Induction For each n e N letAn be a mathematical statement an assertion that is either true or false If a A1 is true and b for each k e N for which Ak is true Ak 1 also is true then An is true for all n e N In international intrigue the PMI is known as the domino theory Pascal s Triangle ab0 1 ab1 ab ab2 a22abb2 ab3 a3 3a2b3ab2 193 ab4 a4 46232 6a2b2 4ab3 194 Pascal s Triangle row 0 1 row 1 1 1 row 2 1 2 1 row 3 1 3 3 1 row 4 1 4 6 4 1 The entries in Pascal s triangle are called binomial coefficients The binomial coefficient in Pascal s triangle in row n and position k from the left is denoted by n k fork O12nandn O123 The binomial coefficients are defined inductively by the pattern in Pascal s triangle Eli ll quotlelxllm If we define 203 we holds fork O12n Binomial Theorem AKA Binomial Expansion FOIL at its best Theorem If a b e R and n e N then ab 1 ankbk k0 You should be able to guess the following rule by adding at most two rows to Pascal s triangle and doing some trial and error calculations n nn 1n 2n k1 k 123k with kfactors in both the numerator and denominator More compactly n k kn k fork O12n Upper Bounds Lower Bounds and Related Matters LetE c R A real numberMis an upper bound lower bound ofE ifx 5 M x 2 M for each x e E A set E c R is bounded above below it has an upper lower bound A set is bounded if it is both bounded above and bounded below A real numberM is the maximum element of a setEifM e Eansz xforeveryx e E Notation M maXE A real number m is the minimum element of a set Eifm e Eandm 5 xforeveryx e E Notation m minE If maXE exists it is the least upper bound of the set If minE exists it is the greatest lower bound of the set What if there is no max or min A real number s is called the supremum ofE or the least upper bound ofE ifs is an upper bound for E and s 5 Mfor any upper bound Mof E A real number 1 is called the infimum ofE or the greatest lower bound of E if t is a lower bound for E and 12 Mfor any lower bound M of E Notation If a set E has a supremum it is denoted by supE If a set E has an infimum it is denoted by infE Basic Facts If a set has an upper bound it has infinitely many If a set has a supremum or maximum it is unique If a set has a supremum it may or it may not belong to the set If supE E E then supE maXE What are the corresponding statements for infima The Key to Working With Suprema Theorem Approximation Property for Suprema If a setE has a supremum supE then for each s gt 0 there is an element x in E such that supE e lt x 5 supE Equivalently if a set E has a supremum supE then for each c lt supE there is an elementx in E such that cltx supE What are the corresponding statements for infima The Completeness Axiom Distinguishing R from Q there are no holes in the real number line Postulate 4 The completeness Axiom Every nonempty set of real numbers that is bounded above has a supremum Important Matters Every nonempty set E that is bounded above has a supremum However the supremum need not belong to the set If a set has a supremum and that supremum belongs to the set we also call the supremum the maximum element of the set Roughly speaking the supremum is a quotsubstitutequot for a maximum element of a set when one does not exist What are the corresponding statements for infima Uniqueness of the Real Number System Postulates 1 2 and 4 characterize the real numbers up to isomorphism Fundamental Consequences of Completeness of R l The Archimedian Property of R There are arbitrarily large positive integers equivalently there are arbitrarily small reciprocals of positive integers Many basic limit arguments rely on these facts Theorem The following three equivalent properties hold in the real number system a N is not bounded above that is given any real number b there is n e N such that n gt b b If b e R and x gt 0 then there is an n e N such that nx gt b c For each s gt 0 there is an n e N such that O lt ln lt s II On Expected Properties of Z and R Expected maxima and minima are there The integers separate the reals as expected R Un Znan Theorem le c Z is nonempty and bounded above then maXE exists What is the corresponding result forE c Z bounded below It implies that the completeness axiom implies the wellordering axiom for N Theorem lfx e R there is a unique n e Z such that n 5 x lt n 1 Remark This n is the greatest integer in x Notation n x lll Q is dense in R Real numbers can be approximated arbitrarily closely by rational numbers Theorem The following equivalent properties hold in the real number system a Given any two real numbers ab with a lt b there is a rational number q such that a lt q lt b b Given x e R and s gt 0 there is a q e Q such that x q lt s c Given x e R and n e N there is a q e Q such that x q lt The General Definition of a Function A function is its graph LetXand Y be sets The Cartesian product of these sets is the set of all ordered pairs of elements taken from X and from Y XxYxyxeXandy Y A function between Xand Y is a nonempty subset ofXx Ysuch that if xy Efand xy Eftheny y I The domain offis the set domm x X Eye YSt xy 6f The range offis the set ranm y e Y 3x EXSt xy 6f The set Y is called the codomain off When domm X we sayfis a function from Xto Yand write f X gt Y Because of I each x e domm determines a uniquey e Ysuch that xy ef This uniquey is denoted by fx and we are back to the more familair notation for a functionf y fx Basic Properties a Function May Have A function f X gt Y is onto or surjective if ranm Y A function f X gt Y is onetoone or injective if xix gtfx i x Equivalently f X gt Y is onetoone if fx fx gtx x A function f X gt Y that is both injective and surjective is bijective If a functionf X gt Y is bijective we can define a new function g Y gt be g yx E YXX xy 6f By definition domg ranm and rang dom and x gy ltgt y fx Equivalently fgy y and gfx x for all x e dom and y e domg Because of these relations g is called the inverse function of f and usually is denoted byf l Realvalued Functions of a Real Variable AfunctionfX gt YwithXc Rand Yc Risa realvalued function of a real variable For such functions we define f X gt Yis increasing if xx E domm and x lt x gtfx fx f X gt Y is decreasing if xx E domm and x lt x gtfx z x f X gt Y is strictly increasing if xx E domm andx lt x gtfx ltfx f X gt Y is strictly decreasing if xx E domm andx lt x gtfx gtfx f is monotone if it is either increasing or decreasing and is strictly monotone if it is either strictly increasing or strictly decreasing Mth 311 November 19 1997 l Abel Dedekind Dirichlet Theorem Recall the alternating series test Theorem 1 Alternating Series Test If 1 12nngt1 is monotone decreasing and 2 11mm 12 0 then the series 00 Z 1 bn n1 COHVEI gES This result is easily proved by grouping terms and using the Cauchy criterion or by considering even and odd partial sums separately and using the fact that bounded monotone sequences converge It is a nice result but it has two drawbacks one that the sign alternate strictly and two the monoticity hypothesis The monoticity even when true is frequently awkward to establish We can not eliminate these hypotheses completely since the theorem is false without them but we can modify them A sequence 12nngt1 is said to be absolutely convergent bad terminology or of bounded variation better terminology if 2 lbn17bnlltoo n Exercise 1 If 12nngt1 is of bounded variation then limneoo bn exists Exercise 2 If 12nngt1 is a bounded monotone sequence increasing or decreasing then 12nngt1 is of ounded variation 7 7 Since we are dealing with series which may fail to be absolutely convergent convergence will depend on delicate cancellation Thus our arguments will proceed by grouping terms appropriately just as we had to do in the proof of the alternating series test The primary tool for these types of arguments is Abel39s summation by parts Consider two sequences awngt1 and bnngt1 Define partial sums An by A0 0 and for n 2 1 7 7 71 An Z ak k1 Note an An 7 Ansl for n 21 Thus for n gt m 2 akbk Z Ak Akil k km1 km1 n n71 Z Akbk Z Akbk1 km1 km Anbn1 Ambm1 2 Ag bk 7 bk1 km1 Bent Petersen File ref 3llabel tex This is the partial summation formula Taking m 0 we have 2 akin Anbw 2 A In 7 m1 k1 k1 which leads immediately to a simple convergence result Lemma 2 If awngt1 and 12nngt1 are sequences ofcompex numbers n An 2 Wm k1 1 the series 221 An bn 7 127 converges and 2 the sequence Anbn1n21 converges then the series COHVEI gES The trick to getting a useful result is to strengthen the hypotheses a bit but not too much so as to get conditions that can frequently be conveniently checked The two main results are theorems of Abel Dedekind and Dirichlet dating from the 186039s Theorem 3 Abel Dedekind Ifthe series 221 an converges and the sequence 12nngt1 is of bounded variation then the series T 00 2 am COHVEI gES Proof If An 221 ak then the sequence 147 gt1 converges Since 12nngt1 is of bounded variation it also converges Thus the sequence Anbn1ngt1 converges Now since Anngt1 converges it is bounded and so there exists a constant M gt 0 such that lAnl g M for each n Then lAnbn bn1lSMlbnibn1l for each n implies the absolute convergence of 221 An bn 7 Inn The theorem now follows from lemma 2 The original theorem of Abel from the 182039s required that the sequence 12nngt1 be monotone and bounded which is a stronger hypothesis than bounded variation T If we weaken the hypothesis on the sequence Anngt1 and strengthen the hypotheses on the sequence 12nngt1 suitably we obtain a very useful result T Theorem 4 Abel Dedekind Dirichlet If the series 221 an has bounded partial sums and the se quence 12nngt1 is of bounded variation and limnsm bn 0 then the series g anbn 711 COHVEI gES Proof If An 221 ak then by hypothesis there is a constant K gt 0 such that l Anl g K for each n It follows that l Anbn1 l S K l bn1 l for each n which implies whirl Anbn1 0 In addition we have lAnbn bn1lSKlbn bn1l for each n which implies the absolute convergence of 221 An 12 7 127 The theorem now follows from lemma 2 D A bit weaker theorem sometimes called the Abel Dirichlet theorem requires 127ngt1 be a monotone sequence converging to 0 7 lfwe take an 71 in the Abel Dedekind Dirichlet theorem we obtain an alternating series test which does not require monoticity Corollary 5 Abel Dirichlet Dedekind alternating series test If 127ngt1 is a sequence of complex numbers 7 1 221 lbn bn1l lt 00 and 2 limmoo b7 0 then the series 00 Z HW n1 converges Since a bounded monotone sequence has bounded variation the corollary is a generalization of the alternating series test While the corollary is frequently used in cases where 12 2 0 so we do actually have alternating sign this is not required Thus we can replace 1 with 71 127 and obtain a result which brings out very clearly the dependence on cancellation Corollary 6 If 127ngt1 is a sequence ofcompex numbers 1 221 lbn bn1l lt 00 and 2 11mm 12 0 then the series M8 i S ll i converges Example 1 From the trigonometric identity Sinltnxgt 7 cosn 7 7 005n 2 sinz2 we obtain 1 Z sinkz S fl g l sumac2 l 3 Thus by the Abel Dedekind Dirichlet theorem 2 bn sinnz 711 converges if 12nngt1 is a sequence with bounded variation and 11mH00 bn 0 for example ifbn Example 2 The sequence 1 ngt1 is bounded and monotone and so of bounded variation Thus oo lfzn1 an converges the so does 00 711 n 1 an n Note this result is not at all exciting if 221 an happens to be absolutely convergent Example 3 The sequence gt n 2 series with bounded partial sums then converges Thus converges is a monotone sequence converging to 0 Thus if 221 an is a i sinnz n2 logn Copyright 1997 by Bent E Petersen document for non profit educational purposes provided that no alterations are made and provided that this copyright notice is preserved on all copies Bent E Petersen Department of Mathematics Oregon State University Corvallis OR 973314605 http ucs orst eduquotpeterseb http www peak org39petersen http web orst eduquotpeterseb Permission is granted to duplicate this phone numbers office 541 7375163 home 541 7531829 fax 541 7370517 bentQalum mit edu petersenemath orst edu 90M 9 5 on F1 MTH 311 Chapter 2 What is a sequence 7 7 Discuss convergence of a geometric sequence De ne a sequence Which has a limit is convergent has a convergent sub sequence is divergent is bounded is lucky is monotone raW enn diagrams Which show Which of the sets of sequences are subsets of Which sets in the universe of all sequenceslh Give an example if possible of o a sequence convergent to 3 o a sequence With three convergent subsequences o a sequence bounded by the numbers 71 and 275 o a sequence Which is NOT convergent o a divergent sequence 0 an unbounded sequence 0 a Cauchy sequence 0 a Cauchy sequence Which is not convergent o a monotone sequence 0 a nonmonotone sequence Which is convergent o a sequence Which is monotone and bounded o a sequence Which is monotone and unbounded o a sequence Which is convergent and unbounded o a sequence Which is bounded and divergent o a sequence Which is bounded and divergent to 007 Formulate the BolzanoWeierstrass Theorem sketch its proof give exam ple of an application 7 Assume limnn00 In a and limnn00 yn 127 What can you say about the sequence un znym 2n nzn wn Eng 7 Prove it lfV n In 2 yn then What can you say about a and b 7 What if we actually know tha Vn zngtyn7 Assume that given sequences In and yn Which are each convergent to the same number a E R that there is a sequence rn such that In S rn S yw What can you say about limnn00 rn 7 Prove it Give example of an application The Derivative at a Point DEF A function f dom gt R is differentiable at a E dom if f x i f a fitl 3 lim 6 1R ran in Which case this limit is the derivative of f at a and is denoted by any of df df 1001 a 7 a sza 7 za a The limit in the de nition also can be expressed by m fah 4a hgt0 h DEF If the domain of f is a union of intervals and if a leftehand endpoint of one of those intervals belongs to that interval7 then differentiability of f at such an endpoint and its derivative there are de ned using a rightehand limit Likewise7 leftehand limits are used to de ne differentiability and the derivative at a rightehand endpoint of an interval in the domain of f that belongs to the domain Th If f is differentiable at a then f is continuous at a The Derivative as a Function DEF Let f domf gt lR Suppose f exists for at least one at E domf The function f With values f and With domain dom fl ac E dom f exists is called the rst derivative of f The derivative of f may it self be differentiable at certain points in its doe main If so we de ne the second derivative of f by f f and so on for higher df dom A R erivatives The nth derivative of f is often denoted by d f fol orby Worby an I Facts 1 A function may not be differentiable at any point in its domain 2 A function may be differentiable once but not tWice at a point in its domain 3 A function may be differentiable any number of times at a point in its domain Big Oh Little Oh What Begins With Oh DEF We say a function f h is O h as h A 0 and often denote f h by O h if the function is Lipschitz continuous at h 0 that is7 there is a 6 gt 0 and M lt 00 such that lfhl S Mlhl for all lhl lt 6 DEF We say a function f h is 0 h as h A 0 and often denote f h by 0 h if the function is de ned on some open interval containing h 07 is continuous at h 07 and satis es if Ml 11m 7 hgt0 lhl FACTS For any a b E R Little oh functions are big oh a01b02 O and 1201 1102 o 201 1102 0 000 and 000 are both little oh functions Similar symbolism is used is h A 1 instead of h A 0 6 The limit in the de nition of little oh can be expressed as f h h gt 0 as h A 0 SIMPPJEOE DEF We say a function is differentiable if it is differentiable on its domain DEF A point a of a subset E C R is an interior point of E if there is a nonempty open interval I such that a E I C E Th Another View of Di erentiable Let f dom A R and a be an interior point of dom Then the following are equivalent 1 f is differentiable at a 2 There is a unique real number m and a little oh function a h such that fah famhah as h A 07 in Which case m f a The Basic Differentiation Rules Th Suppose the functions f and g are differentiable at at and c E R Then the functions f 9 cf7 fg are differentiable at at and f9 r f fav 9 x cf 90 Cf 90 f9 90 f 909 90 f 909 90 lf7 in addition7 g 7E 07 then Th Chain Rule If the function f is differentiable at at and the function g is differentiable at f 7 then 9 o f is differentiable at at and g o f at 9 f 90 f at The Power Rule for Rational Exponents Th Let q E Q Then the function f at is differentiable and f qacq l for all at in the dorn acq l That is7 the power rule is true Whenever it makes sense Th All constant functions are differentiable and have derivative zero Th All polynomial and all rational functions are differentiable Oneesided Derivatives DEF A function f dorn gt R is differentiable from the right at a E dornf if 3 m M 4a zgta aria ER in Which case this limit is called the right derivative of f at a and is denoted by f a It is differentiable from the left at a E dorn if 311m fx fa ER magi x 7 a in Which case this limit is called the left derivative of f at a and is denoted b f a Apparently7 a function f is differentiable at a if and only if it is both differene tiable from the right and differentiable from the left in Which case f a f a fL a

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