AT&T 1234 s 67 3 Plot of a sequence with limit L. FM any the dots eventually an Oband around L. 3:11 PM SECTION S" Our main goal is to study convergence Of A Ian I cmVerges to a limit L if Ian Ll becomes arbitrarily small when n is sufficiently large. Here is the formal definition. DEFINITION Limit of a Sequence We say L and write lim an = L or L if, for every > O, there is a nurntrr M such that fan Ll < Rx all n > M. If no limit exists, we say that diverge. If the terms increase without bound. we say that diverge to infinity. If Ian I converges. then its limit L is unique. To visualize the limit plot the *lints (I. (2. a2). (3, , as in figure 3. TIE sequenceconvergesto L if. O. the plotted points eventually remain within an around the rxyizontal line y L. Figure 4 shows the plot of a sequence converging to L = l. On the other hand. since it continually cycles through l. 0.1.0, cn = in figure 5 has no limit. 10 12 2 4 6 s 10 FIGURE 4 Sequence an = 12 14 FIGURE 5 The equence Cn no limit. . prove that lim an EXAMPLE 3 proving Convergence Let an = Solution The definition requires us to find, for every > O. a M such that for all n > M We have Therefore, Ian if I I < eforalln>l In Other words, Ian 3. This proves that lim an I Note the following two facts about sequences: The limit does not change if we change Or drop finitely many terms of the sequence. If C is a constant and an C for all n greater than some fixed value N, then lim an = C. Many of the sequences we consider are defined by functions; that is. an = f(n) for some function f. For example, is defined by