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Chapter 1: Problem 174

Let \(x_0\) be a rational number, and let \(\left(x_n\right)_{n \geq 0}\) be the sequence defined recursively by $$ x_{n+1}=\left|\frac{2 x_n^3}{3 x_n^2-4}\right| $$ Prove that this sequence converges, and find its limit as a function of \(x_0\).

Short Answer

Expert verified
The sequence converges to 0 regardless of the initial value.

Step by step solution

01

Analyze the recursive formula

Given the sequence definition, rewrite the formula:\[ x_{n+1} = \left| \frac{2 x_n^3}{3 x_n^2 - 4} \right| \]
02

Find the fixed points

A fixed point is a value \(L\) such that \(L = \left| \frac{2 L^3}{3 L^2 - 4} \right|\). Set up the equation and solve for \(L\):\[ L = \left| \frac{2 L^3}{3 L^2 - 4} \right| \] Since we're dealing with absolute value, break it into cases when the inner value is non-negative and negative.
03

Case 1 - Non-negative

Assume \( \frac{2 L^3}{3 L^2 - 4} \geq 0 \), then the equality holds without the absolute value:\[ L = \frac{2 L^3}{3 L^2 - 4} \]\[ L (3 L^2 - 4) = 2 L^3 \]\[ 3 L^3 - 4 L = 2 L^3 \]\[ L^3 - 4 L = 0 \]\[ L (L^2 - 4) = 0 \]\[ L = 0 \text{ or } \pm 2 \]
04

Case 2 - Negative

Assume \( \frac{2 L^3}{3 L^2 - 4} < 0 \), then the equality holds with a negative sign:\[ L = - \frac{2 L^3}{3 L^2 - 4} \]Which simplifies to:\[ L (3 L^2 - 4) = -2 L^3 \]\[ 3 L^3 - 4 L = -2 L^3 \]\[ 5 L^3 - 4 L = 0 \]\[ L (5 L^2 - 4) = 0 \]\[ L = 0 \text{ or } \pm \sqrt{\frac{4}{5}} \]
05

Check for convergence

As \( x_n \to 0 \), \( x_{n+1} = \left| \frac{2 x_n^3}{3 x_n^2 - 4} \right| \) approaches 0 as well.For initial values, since the sequence is defined using absolute values, any positive or negative initial value will ultimately converge to 0, given the recursive nature of the function.
06

Conclusion

Given that both cases support 0 as the limit and the recursive relationship makes the absolute value progressively smaller, the sequence converges to 0 regardless of the initial value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fixed Points
In mathematics, a fixed point of a function is an element of the function's domain that is mapped to itself by the function. To find a fixed point of a recursive sequence, we solve for a value, L, such that: \[ L = \frac{2L^3}{3L^2 - 4} \]
Since the absolute value affects the sign, we consider two cases.
  • Case 1: When \( \frac{2L^3}{3L^2 - 4} \geq 0 \), remove the absolute value and solve the equation normally.
  • Case 2: When \( \frac{2L^3}{3L^2 - 4} \< 0 \), include a negative sign to equate L.
In both situations, by simplifying, we find possible fixed points for our sequence as: 0, 卤2, and 卤鈭�(4/5).
Understanding fixed points helps determine where recursive sequences might stabilize.
Convergence of Sequences
Convergence in a sequence means that as n approaches infinity, the terms of the sequence approach a particular value, known as the limit. For a sequence defined by: \[ x_{n+1} = \frac{2x_n^3}{3x_n^2 - 4} \]
we need to establish whether it converges. Analyzing the sequence, we observe the behavior of the terms.
If \x_n\ is close to 0, as \ goes to infinity, the recursive equation's numerator (\(2x_n^3\)) becomes very small, and the denominator (\(3x_n^2 - 4\)) balances it.
Since \x_n\ ultimately cycles through smaller absolute values before eventually converging to 0, convergence to 0 becomes evident.
Understanding convergence tells us about the long-term behavior of a recursive sequence.
Absolute Value in Recursion
The absolute value in a recursive formula impacts the sequence by ensuring that terms remain non-negative. For the sequence \[ x_{n+1} = \frac{2x_n^3}{3x_n^2 - 4} \]
moving to \[ x_{n+1} = \frac{2|x_n^3|}{3x_n^2 - 4} \],
if \x_n\ is negative, the absolute value converts it to positive, influencing the next term of the sequence.
This guarantees non-negativity over time and simplifies convergence analysis as negative values are not permitted to propagate.
Understanding how absolute values in recursion work helps in grasping the sequence's behavior when initial terms may oscillate or when certain mathematical properties make the outcome of the sequence easier to predict.
Recursive Formulas
A recursive formula provides a way to compute each term of a sequence based on previous terms. For the sequence given by \[ x_{n+1} = \frac{2x_n^3}{3x_n^2 - 4} \]
it defines each term by manipulating \x_n\.
Recursive formulas are essential in defining sequences that build upon their earlier results.
Some key points about recursive formulas:
  • They enable the generation of series and sequences step-by-step.
  • They can reveal complicated long-term behaviors from simple rules.
  • Understanding their structure and behavior is integral to many areas of mathematical research and application.

By working through recursive formulas, we uncover how sequences evolve and predict their limits and fixed points.

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Most popular questions from this chapter

Suppose we start with a Pythagorean triple \((a, b, c)\) of positive integers, that is, positive integers \(a, b, c\) such that \(a^2+b^2=c^2\) and which can therefore be used as the side lengths of a right triangle. Show that it is not possible to have another Pythagorean triple \((b, c, d)\) with the same integers \(b\) and \(c\); that is, show that \(b^2+c^2\) can never be the square of an integer.

A gambling game is played as follows: \(D\) dollar bills are distributed in some manner among \(N\) indistinguishable envelopes, which are then mixed up in a large bag. The player buys random envelopes, one at a time, for one dollar and examines their contents as they are purchased. If the player can buy as many or as few envelopes as desired, and, furthermore, knows the initial distribution of the money, then for what distribution(s) will the player's expected net return be maximized?

Let \(g\) be a continuous function defined on the positive real numbers. Define a sequence \(\left(f_n\right)\) of functions as follows. Let \(f_0(x)=1\), and for \(n \geq 0\) and \(x>0\), let $$ f_{n+1}(x)=\int_1^x f_n(t) g(t) d t $$ Suppose that for all \(x>0, \sum_{n=0}^{\infty} f_n(x)=x\). Find the function \(g\). \(\quad\)

Let \(f\) be a continuous function on \([0,1]\), which is bounded below by 1 , but is not identically 1 . Let \(R\) be the region in the plane given by \(0 \leq x \leq 1\), \(1 \leq y \leq f(x)\). Let $$ R_1=\\{(x, y) \in R \mid y \leq \bar{y}\\} \quad \text { and } \quad R_2=\\{(x, y) \in R \mid y \geq \bar{y}\\} $$ where \(\bar{y}\) is the \(y\)-coordinate of the centroid of \(R\). Can the volume obtained by rotating \(R_1\) about the \(x\)-axis equal that obtained by rotating \(R_2\) about the \(x\)-axis?

The proprietor of the Wohascum Puzzle, Game, and Computer Den has invented a new two-person game, in which players take turns coloring edges of a cube. Three colors (red, green, and yellow) are available. The cube starts off with all edges uncolored; once an edge is colored, it cannot be colored again. Two edges with a common vertex are not allowed to have the same color. The last player to be able to color an edge wins the game. a. Given best play on both sides, should the first or the second player win? What is the winning strategy? b. There are twelve edges in all, so a game can last at most twelve turns (whether or not the players use optimal strategies); it is not hard to see that twelve turns are possible. How many twelve-turn end positions are essentially different? (Two positions are considered essentially the same if one can be obtained from the other by rotating the cube.) \(\quad\)
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