Download SEQUENTIAL LIMITS Sequential Limits Background : start with a

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)
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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?
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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?
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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).
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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.
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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.