Proving Boundedness of Recursive Sequences: A Mathematical Approach

Understanding the behavior of recursive sequences is essential in various fields, including mathematics and computer science. In this article, we will delve into the mechanics of proving the boundedness of a specific recursive sequence using a mathematical approach and mathematical induction. Specifically, we will explore the sequence defined by the recurrence relation:

an 1 4 - frac{7}{2a_{n-1}}

Introduction to the Sequence

Consider the recursive sequence defined by the formula:

To establish the boundedness of this sequence, it is crucial to first define the base case. Let us assume 1 le a_1 le 3. This assumption is necessary because it will help us to prove that the sequence is both bounded below by 1 and bounded above by 3.

Base Case and Inductive Hypothesis

The first step in proving the boundedness is to check the base case:

1 a_1 le 3

Now, let's assume that for some integer k 1, the following inequalities hold:

1 le a_k le 3

We aim to show that these inequalities also hold for a_{k 1}:

a_{k 1} 4 - frac{7}{2a_{k-1}}

Inductive Step

First, multiply both sides by a_{k-1} to get:

a_{k 1} a_{k-1} 4 a_{k-1} - frac{7}{2}

Subtracting a_{k 1} from both sides, we obtain:

a_{k 1} a_{k-1} - a_{k 1} 4 a_{k-1} - a_{k 1} - frac{7}{2}

Rearrange to factor:

a_{k 1} (a_{k-1} - 1) 4a_{k-1} - frac{7}{2}

Divide by (a_{k-1} - 1) (noting that a_{k-1} eq 1), we get:

a_{k 1} frac{4a_{k-1} - frac{7}{2}}{a_{k-1} - 1}

Using the assumption 1 le a_k le 3, we can verify:

Lower Bound

1 le a_{k 1} le 3 Rightarrow 1 le a_k le 3 Rightarrow 4 - frac{7}{3} le a_{k 1} le 4 - frac{7}{6}

This simplifies to:

1 le a_{k 1} le 3

Upper Bound

1 le a_{k 1} le 3 Rightarrow 4 - frac{7}{2a_k} le 3 Rightarrow 1 le a_{k 1} le 3 Rightarrow frac{8a_k - 3}{2a_k 1} le 3

This simplifies to:

1 le a_{k 1} le 3

Conclusion

By mathematical induction, we have shown that if 1 le a_1 le 3, then the sequence a_n is bounded both below by 1 and above by 3. This is a crucial property for understanding the behavior of the sequence over time and for various applications in areas such as numerical analysis and algorithm design.

Related Topics

Further investigation into the behavior of this sequence can lead to a deeper understanding of its properties. For example, examining the fixed points and the stability of the sequence can provide insights into its long-term behavior. Additionally, exploring the general recurrence relation:

a_{n 1} frac{8a_n - 3}{2a_n 1}

Can uncover more complex patterns and properties, such as the convergence of the sequence to a fixed point or the periodicity of the sequence.

Key takeaways from this article include:

The importance of establishing a base case for induction. The use of mathematical induction to prove the boundedness of sequences. The concept of fixed points and their role in the convergence of sequences. The significance of understanding the stability of sequences for practical applications.

By delving into these areas, one can gain a comprehensive understanding of how to analyze and work with recursive sequences, a fundamental concept in mathematics and beyond.