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# ANALYSIS I MATH 554

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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.

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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

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