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DEFINITION OF A SEQUENCE A sequence is a set of numbers u1; u2; u3; . . . in a deﬁnite order of arrangement (i.e., a correspondence with the natural numbers) and formed according to a deﬁnite rule. Each number in the sequence is called a term; unis called the nth term. The sequence is called ﬁnite or inﬁnite according as there are or are not a ﬁnite number of terms. The sequence u1; u2; u3; . . . is also designated brieﬂy by fung. 1. The set of numbers 2; 7; 12; 17; . . . ; 32 is a ﬁnite sequence; the EXAMPLES. un¼ 2 5ًn 1 ¼ ق5n nth term is given by 3, n ¼ 1; 2; . . . ; 7. 2. The set of numbers 1; 1=3; 1=5; 1=7; . . . is an inﬁnite sequence with n ¼ 1; 2; 3; . . . . nth term un¼ 1=ً2n 1ق, Unless otherwise speciﬁed, we shall consider inﬁnite sequences only. LIMIT OF A SEQUENCE A number l is called the limit of an inﬁnite sequence u1; u2; u3; . . . if for any positive number ﬁnd a positive number N depending on we can such that ju lj < for all integers n > N. In such case we n write lim un¼ l. n!1 EXAMPLE . If un¼ 3 1=n ¼ 3n 1=قn, the sequence is 4; 7=2; 10=3; . . . and we can show that lim u n¼ 3. n!1 If the limit of a sequence exists, the sequence is called convergent; otherwise, it is called divergent. A sequence can converge to only one limit, i.e., if a limit exists, it is unique. See Problem 2.8. A more intuitive but unrigorous way of expressing this concept of limit is to say that a sequence u1; u2; u3; . . . has a limit l if the successive terms get ‘‘closer and closer’’ to l. This is often used to provide a ‘‘guess’’ as to the value of the limit, after which the deﬁnition is applied to see if the guess is really correct. THEOREMS ON LIMITS OF SEQUENCES If lim an¼ A and lim n!1 1. lim ًan bn¼ ق n!1 2. lim an lim bn¼ A B n!1 lim ًanbn¼ ق n!1 3. bn¼ B, then n!1 n!1 lim anlim n!1 bn¼ A lim ًanbn ¼ قlim an قlim n!1 n!1 B n!1 bn ¼ قAB n!1 23 Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. 24 SEQUENCES 4. lim an A ¼ lim an¼ n!1 if lim n!1bn B n!1 bn lim bn [CHAP. 2 ¼B6¼ 0 n!1 an does not exist. If B ¼ 0 and A 6¼ 0, lim n!1 bn If B ¼ 0 and A ¼ 0, lim 5. 6. n!1 bnmay lim ap n!1 an n¼ lim n!1 lim pa ¼ plim n!1 n an قp¼Ap, or may not exist. for p ¼ any real number if Apexists. an ¼ pA, for p ¼ any real number if p Aexists. n!1 INFINITY We write lim an¼ 1 if for each positive number M we can ﬁnd a positive number N (depending on an!1 M) such thatn> M for all n > N. Similarly, we write lim an¼ 1 if for each positive number M we n!1 can ﬁnd a positive number N such that an< M for all n > N. It should be emphasized that 1 and 1 are not numbers and the sequences are not convergent. The terminology employed merely indicates that the sequences diverge in a certain manner. That is, no matter how large a number in absolute value that one chooses there is an n such that the absolute value of a is greater than that n quantity. BOUNDED, MONOTONIC SEQUENCES If u @ M for n ¼ 1; 2; 3; . . . ; where M is a constant (independent of n), we say that the sequence n fu g is bounded above and M is called an upper bound. If u A m, the sequence is bounded below and m is n n called a lower bound. If m @ u @ M the sequence is called n bounded. Often this is indicated by ju j @ P. n Every convergent sequence is bounded, but the converse is not necessarily true. If un1Aunthe sequence is called monotonic increasing; if un1>unit is called strictly increasing. Similarly, if u [email protected] the sequence is called monotonic decreasing, while if n u 1<u it is strictly n n n decreasing. EXAMPLES. 1. The sequence 1; 1:1; 1:11; 1:111; . . . increasing. 2. The sequence 1; 3. The sequence 1; 1; 1; is bounded and monotonic increasing. It is also strictly 1; 1; . . . is bounded but not monotonic increasing or decreasing. 1:5; 2; 2:5; 3; . . . is monotonic decreasing and not bounded. However, it is bounded above. The following theorem is fundamental and is related to the Bolzano–Weierstrass theorem (Chapter 1, Page 6) which is proved in Problem 2.23. Theorem. Every bounded monotonic (increasing or decreasing) sequence has a limit. LEAST UPPER BOUND AND GREATEST LOWER BOUND OF A SEQUENCE A number M is called the least upper bound (l.u.b.) of the sequence fu g if u @ M, n ¼ 1; 2; 3; . . . while at least one term is greater than M n n for any > 0. A number m is called the greatest lower bound (g.l.b.) of the sequence fu g if u A m, n ¼ 1; 2; 3; . . . n n while at least one term is less than m for any > 0. Compare with the deﬁnition of l.u.b. and g.l.b. for sets of numbers in general (see Page 6