Limit Of A Sequence

Definition 3.2

A sequence is said lo have the limit L if, for any e >0. however small, there is some value N such that |«„ — L\ < e whenever n > N. Such a sequence is said to be convergent, and we write its limit as lini,,-.^ a„ ~ L.

In less formal language, the definition above states that a sequence has a limit L provided that all values of the sequence "beyond some term" can be made as close to L as one wishes (i.e., the condition \a„ - L\<t can be met for as small a positive number i as one likes by choosing a sufficiently large value of N). For esample, consider the sequence a„ — \/n (figure 3.2), which has die limit L = 0. We see that |a„ - 0| <0.01 for any choice of N > KM), while \an - 0| <0.002 for any choice of N > 500. More generally. \\/n — 0|<-<r requires a choice of /V > l/f. One can think of N as formally being a function of t and so write N(€).

As is also the case for functions in general (see chapter 2), a sequence may be hounded or unbounded. In particular, wc say that a sequence is bounded if there is some finite value K > 0 such that for some N it follows that a„ < K for all /; > /V bounded above and a„ > - K for all n > N bounded below and is not bounded if one or both of these conditions fails to hold. For example, the sequence f(n) = 2n, « = 1.2,3..,, illustrated in figure 3.1 is unbounded because it is not bounded above, while the sequence

illustrated in figure 3.5 is unbounded because it is nor bounded below. A sequence may also be unbounded because it is neither bounded above nor below, as in the sequence

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