1549901369-Elements_of_Real_Analysis__Denlinger_

2.5 Monotone Sequences 91 Theorem 2.5.3 (Monotone Con.vergence Theorem) Every bounded mono-lflf.WI tone sequence converges. More ...

92 Chapter 2 • Sequences We can use sequences to make this meaning precise. To keep it simple, we shall consider only the case w ...

2.5 Monotone Sequences 93 Then, 0 :::; 100 [x - K.d 1 ] < 10. We repeat the above process: :3 unique integer d2 E {1, 2, · · ...

94 Chapter 2 • Sequences Thus, by Theorem 2.3.5, lim O.d 1 d 2 · · · dn = 1. Since a sequence cannot n-+oo have more then one li ...

2.5 Monotone Sequences 95 Part 2- find lim Xn· By P art 1, we know that 3 L = lim X n E R We n-+oo n-+ao proceed to find L. Cons ...

96 Chapter 2 • Sequences (iii) there is one more term than in the expansion of ( 1 + ~) n ( 1 )n+l Putting (i)- (iii) together, ...

2.5 Monotone Sequences 97 Theorem 2.5.11 (A Sequence Converging to .y'(i) Let a be any positive real number. D efine the sequenc ...

98 Chapter 2 • Sequences That is, Xn - Xn+l 2 0, from which it follows that Xn+l :S Xn· (c) By (a) and (b) together, {xn};;:"= 2 ...

2.5 Monotone Sequences 99 It is reasonable to take x 1 = 2 as our first guess. Using a calculator, we find n Xn x2 n 1 2 4 ( 5) ...

100 Chapter 2 • Sequences 1 Then neither an nor u+-is a lower bound for A, so 3an+l EA 3 n u < an+l < min{an,u+ - 1 -}. n+ ...

2.5 Monotone Sequences 101 limit. To analyze a sequence { Xn} for convergence, students are encouraged to calculate a number of ...

102 Chapter 2 • Sequences n 1 Proof. \:/n E N, let Sn = "°"' ~k -. We want to prove that n-oo lim Sn = +oo. k=l Now, \:/n EN, Sn ...

2.5 Monotone Sequences 103 SUMMARY: (a) To the calculator, every convergent sequence is eventually con- stant! (b) To the calcul ...

104 Chapter 2 • Sequences {an} and {bn} of left and right endpoints of the intervals In. Since the intervals are nested, we have ...

2.5 Monotone Sequences 105 Which of the following sequences are eventually monotone (or strictly) in- creasing (or decreasing)? ...

106 Chapter 2 • Sequences 11. Consider the sequence {xn} defined inductively by x 1 = 1, and \:Jn E N, Xn+i = ~Xn + 6. Prove tha ...

2.6 Subsequences and Cluster Points 107 ( c) If { Xn} and {yn} are sequences of nonnegative real numbers that are monotone incre ...

108 Chapter 2 • Sequences The following lemma is simply a technical observation that frequently proves useful in what is to come ...

2.6 Subsequences and Cluster Points 109 Proof. (a) This is merely a restatement of the definition of Xn---+ L. (b) Part 1 (=?):S ...

110 Chapter 2 • Sequences Therefore, Xnk ---+ L. Part 2 ( {::::): Exercise 3. • Example 2.6.9 Find lim (1 + 2 1 ) n n---+cx::i n ...