1. Let f be a continuous function, and define a sequence by a₁ = a, an+1 = f(an). This is called an iterated sequence of f, since we perform the same operation over and over to generate the sequence elements. (a) If limn→∞ an = L show that f(L) L. (We call a value x where f(x) = x a fixed point of f, so another way to say this is: the limit of any convergent iterated sequence of f is a fixed point of f.) = (b) Let g(x) = x³ – x. Find all fixed points of g. (c) Consider the iterated sequences of g: determine which initial values a lead to convergent sequences and which divergent. For the convergent values of a determine their limits. (Hint: think of the fixed points of g like equilibria in a differential equation: some are stable and some unstable.)
1. Let f be a continuous function, and define a sequence by a₁ = a, an+1 = f(an). This is called an iterated sequence of f, since we perform the same operation over and over to generate the sequence elements. (a) If limn→∞ an = L show that f(L) L. (We call a value x where f(x) = x a fixed point of f, so another way to say this is: the limit of any convergent iterated sequence of f is a fixed point of f.) = (b) Let g(x) = x³ – x. Find all fixed points of g. (c) Consider the iterated sequences of g: determine which initial values a lead to convergent sequences and which divergent. For the convergent values of a determine their limits. (Hint: think of the fixed points of g like equilibria in a differential equation: some are stable and some unstable.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. Let f be a continuous function, and define a sequence by a₁ = a, an+1 = f(an). This is called an
iterated sequence of f, since we perform the same operation over and over to generate the sequence
elements.
(a) If limn→∞ an = L show that f(L)
L. (We call a value x where f(x) = x a fixed point of
f, so another way to say this is: the limit of any convergent iterated sequence of f is a fixed
point of f.)
=
(b) Let g(x) = x³ – x. Find all fixed points of g.
(c) Consider the iterated sequences of g: determine which initial values a lead to convergent
sequences and which divergent. For the convergent values of a determine their limits. (Hint:
think of the fixed points of g like equilibria in a differential equation: some are stable and
some unstable.)
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