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SEQUENTIAL LIMITS Sequential Limits Background : start with a sequence a1, a2, a3, . . ., • Sequential Limit Definitions: a) limn→∞ an = L, if an can be made arbitrarily close to L as n becomes large; b) limn→∞ an = ∞, if an can be made arbitrarily large as n becomes large; b) limn→∞ an = −∞, if −an can be made arbitrarily large as n becomes large. • Examples: find limits for the following sequences. a) an = (n2 − 1)/(1 + 2n2) b) an = (1 + 2en/2)/(en/2 − 5) c) an = (1 + 2en)/(en/2 − 5) d) an = sin(5n) 2 SEQUENTIAL LIMITS Examples continued: find limits and how large n should be for |an − L| < .01 a)’ an = (n2 − 1)/(1 + 2n2) b)’ an = (1 + 2en/2)/(en/2 − 5) Sequences and Difference Equations : • Problem: given a difference equation an+1 = f (an), find limn→∞ an. Note: if limit a exists, a is an equilibrium point, with a = f (a). Examples: a) f (x) = x/3, with a1 = 1; b) f (x) = x2, with a1 = .5, a1 = 1.5? 3 SEQUENTIAL LIMITS • Solution methods for difference equations: i) compute some terms and estimate limit, possibly using cobweb graph, and/or ii) use algebra to find equilibrium by solving a = f (a), and/or iii) use algebra to find an = g(n) for some function g(n). Examples: a) f (x) = x/3, with a1 = 1; b) f (x) = x2, with a1 = .5, a1 = 1.5? 4 SEQUENTIAL LIMITS • Application Example: discrete logistic model: an+1 = an + ran(1 − an/K). Try r = 1/2, K = 100, a1 = 2. Sequence is 2.98, 4.42, 6.54, 9.60, 13.9, 19.9, 27.9, 38.0, 49.7, 62.2, 74.0, 83.6, 90.5, 94.8, 97.3, 98.6, 99.3, 99.6, 99.8, 99.9 ... Equilibrium? • Dominated convergence theorem for an+1 = f (an): if a continuous f (x) is increasing for x ∈ [a, b] and [f (a), f (b)] ⊂ [a, b], then the sequence {an} is either increasing or decreasing and limn→∞ an = a = f (a). 5 SEQUENTIAL LIMITS • More Application Examples: b) Beaverton-Holt Fisheries Model (2.5.38): for population abundance Nn+1 = rNn/(1 + cNn), with r > 0, c > 0, N1 = 1. Determine first four terms; determine equilibria; determine conditions for extinction vs. persistence. 6 SEQUENTIAL LIMITS b) Fibonacci Sequence (1.7.38): for rabbit pairs Rn+1 = Rn + Rn−1. Consider an = Rn/Rn−1 and find limn→∞ an.