Problem 8
Question
Danger: This problem may scramble and fry your brain. Consider the sequence
defined by,
$$
Q_{0}=0.2, \quad Q_{t+1}=Q_{t}+2 Q_{t}\left(1-Q_{t}\right) \quad t=1,2, \cdots
$$
Show that the subsequence \(Q_{2}, Q_{4}, Q_{6}, \cdots\) converges to \(1 .\) It
certainly appears so from the data in Table \(14.6 .6,\) but even
\(Q_{100,000}=1.001114938\) is 0.001 above
1\. Assume without proof that all of the numbers \(Q_{2 t}\) satisfy \(1
Step-by-Step Solution
Verified Answer
The sequence \( Q_{2t} \) converges to 1 as shown via decreasing and bounded properties with a polynomial \( H(x) \).
1Step 1: Function Composition
Start by examining the given function for the sequence \[ G(x) = F(F(x)) = -8x^4 + 24x^3 - 24x^2 + 9x \]This is the function resulting from composing the iteration function \( F(x) = 3x - 2x^2 \) with itself. This composition is critical for understanding how the subsequence \( Q_{2t} \) evolves.
2Step 2: Derive the Relationship
Show the equation\[ Q_{2t+2} - 1 = (Q_{2t} - 1)(-8Q_{2t}^3 + 16Q_{2t}^2 - 8Q_{2t} + 1) \]The form \((Q_{2t} - 1)H(Q_{2t})\) indicates that the evolution depends on both the deviation \( Q_{2t} - 1 \) from 1 and the behavior of the polynomial \( H(x) = -8x^3 + 16x^2 - 8x + 1 \).
3Step 3: Analyze the Polynomial Behavior
Plot \( H(x) \) and observe its characteristics over the interval [0.5, 1.2]. The highest point of \( H \) within this interval is at \((1,1)\), and the function stays positive across this range. This positivity is key in showing that \( Q_{2t} \) converges towards 1 by demonstrating \( H(x) < 1 \) for \( x > 1 \).
4Step 4: Prove Decrease in Sequence
Using positivity and the bounded property of \( H(x) \), you can show\[ Q_{2t+2} - 1 < Q_{2t} - 1 \]This demonstrates the decreasing nature of the sequence towards 1, as the second factor in the equation is less than 1.
5Step 5: Establish Boundedness and Limit
By proving \( Q_{2t+2} - 1 < Q_{2t} - 1 \), and knowing the sequence is bounded above 1 initially, use Theorem 5.2.1. This theorem states that a bounded non-decreasing sequence converges; hence the sequence approaches a limit \( L \leq 1 \).
6Step 6: Find the Limit
Given that \( H(L) \geq H(Q_{2t}) \) for all \( t \), and since the chain inequality\[ Q_{2t} - 1 \leq (Q_{2} - 1)(H(L))^t \]applies, explore \( L \) whether 1 or \( L > 1 \). If \( L = 1 \), the assertion is true. If \( L > 1 \), continue by demonstrating \[ \lim_{t \rightarrow \infty}(Q_{2t} - 1)(H(L))^t = 0 \]Ultimately proving that \[ \lim_{t \rightarrow \infty}Q_{2t} = 1 \]
Key Concepts
Sequence ConvergenceFunction CompositionDynamic SystemsPolynomial Behavior
Sequence Convergence
In calculus, sequence convergence is an important concept that describes the behavior of sequences as they progress towards a specific value. In this particular problem, we examine the sequence \(Q_{2}, Q_{4}, Q_{6}, \cdots\) and aim to show that it converges to 1. Convergence means that as the terms increase, they get closer and closer to a certain number, which in our case is 1.
Sequence convergence relies on two main ideas: boundedness and monotonicity. A bounded sequence is limited to a certain set of values, and a monotonic sequence consistently increases or decreases. When a sequence has these two properties, it is said to have a limit. Here, our sequence is both bounded and decreasing, allowing us to apply theorems to affirm convergence. This means that even though the terms \(Q_{2 t+2}\) seem to decrease slightly without approaching exactly 1, they indeed approach 1 infinitely close as \(t\) becomes very large.
Sequence convergence relies on two main ideas: boundedness and monotonicity. A bounded sequence is limited to a certain set of values, and a monotonic sequence consistently increases or decreases. When a sequence has these two properties, it is said to have a limit. Here, our sequence is both bounded and decreasing, allowing us to apply theorems to affirm convergence. This means that even though the terms \(Q_{2 t+2}\) seem to decrease slightly without approaching exactly 1, they indeed approach 1 infinitely close as \(t\) becomes very large.
Function Composition
Function composition is when one function is applied to the result of another, creating a new function. Here, we look at the function \(G(x) = F(F(x)) = -8x^4 + 24x^3 - 24x^2 + 9x\), which is derived by composing the function \(F(x) = 3x - 2x^2\) with itself.
This composition helps us understand how the sequence \(Q_{2t}\) evolves. By using composition, we can analyze how the sequence changes at every specified interval (every second term for \(Q_{2t}\), \(Q_{4t}\), etc.). This allows us to study the long-term behavior of the sequence, essential for proving convergence. Function composition simplifies complex behaviors by combining them into a single function, thus making the problem more manageable.
This composition helps us understand how the sequence \(Q_{2t}\) evolves. By using composition, we can analyze how the sequence changes at every specified interval (every second term for \(Q_{2t}\), \(Q_{4t}\), etc.). This allows us to study the long-term behavior of the sequence, essential for proving convergence. Function composition simplifies complex behaviors by combining them into a single function, thus making the problem more manageable.
Dynamic Systems
Dynamic systems in calculus refer to mathematical models used to describe how complex systems change over time. These systems often involve sequences or recursive equations like the one in this problem. Here, the iteration defined by the function \(G(x)\) applies repeatedly to understand the evolution of the sequence components.
The dynamic nature of this setup means that each term's value influences the next, creating a feedback loop that self-corrects over the progression of terms. Such systems are instrumental in understanding how small changes can affect the entire sequence's future behavior. This exercise illustrates a typical dynamic system where each iteration brings the sequence closer to a steady state or limit—in this case, convergence to 1. Recognizing and working with dynamic systems aids in predicting and analyzing long-term behavior within calculus problems.
The dynamic nature of this setup means that each term's value influences the next, creating a feedback loop that self-corrects over the progression of terms. Such systems are instrumental in understanding how small changes can affect the entire sequence's future behavior. This exercise illustrates a typical dynamic system where each iteration brings the sequence closer to a steady state or limit—in this case, convergence to 1. Recognizing and working with dynamic systems aids in predicting and analyzing long-term behavior within calculus problems.
Polynomial Behavior
Polynomials express relationships among variables using coefficients and powers. In our case, we analyze the polynomial \(H(x) = -8x^3 + 16x^2 - 8x + 1\). This polynomial plays a key role in showing that the sequence \(Q_{2t}\) approaches 1.
Understanding polynomial behavior is crucial. By evaluating \(H(x)\) over a specific interval [0.5, 1.2], we see that it reaches a peak at (1,1) and remains positive. This behavior implies that as \(Q_{2t}\) gets closer to 1, the output of \(H(x)\) ensures that each subsequent term is even closer than the last. The positivity of \(H(x)\) confirms our conclusion about sequence convergence.
Studying polynomial expressions like \(H(x)\) in the context of limits and convergence allows us to explore how slight changes in the sequence affect its progression over time. Polynomials often serve as central tools for such analyses in calculus by elucidating the critical features impacting system behavior.
Understanding polynomial behavior is crucial. By evaluating \(H(x)\) over a specific interval [0.5, 1.2], we see that it reaches a peak at (1,1) and remains positive. This behavior implies that as \(Q_{2t}\) gets closer to 1, the output of \(H(x)\) ensures that each subsequent term is even closer than the last. The positivity of \(H(x)\) confirms our conclusion about sequence convergence.
Studying polynomial expressions like \(H(x)\) in the context of limits and convergence allows us to explore how slight changes in the sequence affect its progression over time. Polynomials often serve as central tools for such analyses in calculus by elucidating the critical features impacting system behavior.
Other exercises in this chapter
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