Problem 18 - Advanced Calculus

Question

Let \(x_{1}=1, x_{2}=3\), and define all later terms recursively by \(x_{n}=\left(x_{n-1}+x_{n-2}\right) / 2\). Thus, \(x_{3}=2, x_{4}=5 / 2 .\) Is the sequence \(\left\\{x_{n}\right\\}\) monotonic? Does it converge?

Step-by-Step Solution

Answer

The sequence is neither entirely increasing nor decreasing, but from the third term onwards, the sequence is monotonically decreasing. The sequence is also bounded. Therefore, the sequence does converge.

Step 1: Initial analysis

First, write down the first few terms of the sequence using the provided recursive formula. We already have the first four terms: 1, 3, 2, 5/2. This sequence neither looks strictly increasing nor decreasing.

Step 2: Subtraction expression

Find the subtraction of two consecutive terms to observe some pattern or apply mathematical induction. The subtraction of two consecutive terms is defined by \(x_{n+1}-x_{n} = \left(x_{n}+x_{n-1}\right) / 2 - x_{n} = (x_{n-1} - x_{n}) / 2 = - (x_{n} - x_{n-1}) / 2\)

Step 3: Mathematical Induction

Mathematical Induction needs to be applied to prove that the sequence is decreasing (hence, monotonic) from the third term onwards. Having \(x_{3}>x_{4}\), we assume this holds for n and need to prove for n+1: \(x_{n+1}-x_{n+2}\). Since \(x_{n+1}>x_{n+2}\), so \(x_{n}

Step 4: Convergence

A sequence converges if and only if the sequence is bounded and monotonic. We've shown that the series is decreasing from n=3, so the sequence is monotonically decreasing from n=3. Also, since \(1 \leq x_{n} \leq 3\), for all n>=1, the sequence is bounded. Therefore, the sequence does converge.

Key Concepts

Recursive Sequences

Recursive sequences are sequences in which each term is defined based on its preceding terms. This means, starting from known initial values, each subsequent term arises from applying a specific formula to previous terms. In our problem, the sequence starts with the values given as \(x_1 = 1\) and \(x_2 = 3\). From there, we generate later terms using the recursive relation \(x_{n} = (x_{n-1} + x_{n-2}) / 2\).

For example, to find \(x_3\), we use the formula and plug in \(x_2\) and \(x_1\):
- \(x_3 = (x_2 + x_1)/2 = (3 + 1)/2 = 2\).
Recurrence relations like these help define sequences with more complex patterns or behaviors.

Understanding recursive sequences helps because they provide a foundation for understanding complex dynamic systems that make decisions based on prior states. They often appear in computer algorithms, financial modeling, and scientific simulations because they succinctly describe processes that "build upon the past."

Monotonic Sequences

Monotonic sequences are sequences that either never increase or never decrease. Such sequences can be categorized as:

• Monotonically increasing sequence: Each term is greater than or equal to the preceding one.
• Monotonically decreasing sequence: Each term is less than or equal to the preceding one.

In the given problem, after evaluating the initial terms, we established that the sequence \(x_n\) neither constantly increases nor decreases initially. However, by applying induction, it becomes evident that from the third term onward, the sequence is decreasing. This is confirmed by showing \(x_{n+1} - x_n = (x_{n-1} - x_n)/2 = - (x_n - x_{n-1})/2\).

Through this relationship, we deduce that the subtraction between consecutive terms remains consistently negative, leading to the conclusion of a decreasing sequence from the third term. Recognizing monotonic patterns is crucial because it helps in analyzing the behavior of sequences, a necessary step in determining convergence.

Mathematical Induction

Mathematical induction is a powerful proof technique often used to establish the truth of an infinite number of statements. It functions in two primary steps:

• Base Case: Prove that a statement holds true for an initial value, typically \(n=1\).
• Inductive Step: Assume the statement is true for some arbitrary \(n=k\), and then prove it also holds true for \(n=k+1\).

In our sequence problem, we utilized mathematical induction to demonstrate that the sequence is decreasing from the third term onward. Initially confirmed with the base case, \(x_3 > x_4\), we assumed it holds for \(n=k\), where \(x_k > x_{k+1}\).

Then, we showed that \(x_{k+1} > x_{k+2}\) by maintaining the relation derived, thereby proving by induction that the sequence continues to decrease. This methodical approach is integral in mathematics to confirm the properties of sequences and series, among many other applications.

Inductive reasoning allows mathematicians to establish truths beyond initial examples, ensuring broader applications and insights into algebraic concepts.

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Problem 17

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Problem 18

Other exercises in this chapter

Problem 17

Let \(a_{n}=\frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2 \cdot 4 \cdot 6 \cdots 2 n}\) (a) Prove that \(\left\\{a_{n}\right\\}\) is convergent. (b) Can you determin

View solution

Problem 17

(a) Is the interior of a connected set necessarily connected? (b)$ Is the closure of a connected set necessarily connected?

View solution

Problem 18

(a) Show that any two disjoint nonempty open sets are mutually separated. (b) Show that any two disjoint nonempty closed sets are mutually separated.

View solution

Problem 19

Using the order properties (iv) to (vii) for the real field, derive the following additional properties: (a) For any \(a, b \in \mathbf{R}\), if \(a

View solution

Key Concepts

Recursive Sequences

Recursive sequences are sequences in which each term is defined based on its preceding terms. This means, starting from known initial values, each subsequent term arises from applying a specific formula to previous terms. In our problem, the sequence starts with the values given as \(x_1 = 1\) and \(x_2 = 3\). From there, we generate later terms using the recursive relation \(x_{n} = (x_{n-1} + x_{n-2}) / 2\).

For example, to find \(x_3\), we use the formula and plug in \(x_2\) and \(x_1\):
- \(x_3 = (x_2 + x_1)/2 = (3 + 1)/2 = 2\).
Recurrence relations like these help define sequences with more complex patterns or behaviors.

Understanding recursive sequences helps because they provide a foundation for understanding complex dynamic systems that make decisions based on prior states. They often appear in computer algorithms, financial modeling, and scientific simulations because they succinctly describe processes that "build upon the past."

Monotonic Sequences

Monotonic sequences are sequences that either never increase or never decrease. Such sequences can be categorized as:

• Monotonically increasing sequence: Each term is greater than or equal to the preceding one.
• Monotonically decreasing sequence: Each term is less than or equal to the preceding one.

In the given problem, after evaluating the initial terms, we established that the sequence \(x_n\) neither constantly increases nor decreases initially. However, by applying induction, it becomes evident that from the third term onward, the sequence is decreasing. This is confirmed by showing \(x_{n+1} - x_n = (x_{n-1} - x_n)/2 = - (x_n - x_{n-1})/2\).

Through this relationship, we deduce that the subtraction between consecutive terms remains consistently negative, leading to the conclusion of a decreasing sequence from the third term. Recognizing monotonic patterns is crucial because it helps in analyzing the behavior of sequences, a necessary step in determining convergence.

Mathematical Induction

Mathematical induction is a powerful proof technique often used to establish the truth of an infinite number of statements. It functions in two primary steps:

• Base Case: Prove that a statement holds true for an initial value, typically \(n=1\).
• Inductive Step: Assume the statement is true for some arbitrary \(n=k\), and then prove it also holds true for \(n=k+1\).

In our sequence problem, we utilized mathematical induction to demonstrate that the sequence is decreasing from the third term onward. Initially confirmed with the base case, \(x_3 > x_4\), we assumed it holds for \(n=k\), where \(x_k > x_{k+1}\).

Then, we showed that \(x_{k+1} > x_{k+2}\) by maintaining the relation derived, thereby proving by induction that the sequence continues to decrease. This methodical approach is integral in mathematics to confirm the properties of sequences and series, among many other applications.

Inductive reasoning allows mathematicians to establish truths beyond initial examples, ensuring broader applications and insights into algebraic concepts.