Problem 12 - Advanced Calculus

Question

For \(c>0,\) consider the quadratic equation $$x^{2}-x-c=0, \quad x>0$$ Define the sequence \(\left\\{x_{n}\right\\}\) recursively by fixing \(x_{1}>0\) and then, if \(n\) is an index for which \(x_{n}\) has been defined, defining$$x_{n+1}=\sqrt{c+x_{n}}$$ Prove that the sequence \(\left\\{x_{n}\right\\}\) converges monotonically to the solution of the above equation.

Step-by-Step Solution

Answer

The sequence \(\{x_n\}\) is monotonically increasing and bounded above. By the Monotone Convergence Theorem, it converges to a limit, which is the positive solution of the quadratic equation \(x^{2} - x - c = 0\), \(x > 0\).

Step 1: Understand the recursive relation

The sequence \(\{x_n\}\) is given by \(x_1 > 0\) and \(x_{n+1} = \sqrt{c + x_n}\) for \(n \geq 1\). We will compare \(x_n\) and \(x_{n+1}\) to show that the sequence is monotonically increasing, i.e., \(x_{n+1} \geq x_n\) for all \(n\). We also know that the limit of the sequence, if it exists, must be a fixed point of the recursive relation, meaning if \(x\) is the limit, then \(x = \sqrt{c + x}\).

Step 2: Show that the sequence is monotonically increasing

Assuming that \(x_{n+1} \geq x_n\) holds for some \(n\), we show that \(x_{n+2} \geq x_{n+1}\). This assumption implies \(x_{n+1} = \sqrt{c + x_n} \geq \sqrt{c + x_{n-1}} = x_n\) for \(n \geq 2\), because the square root function is monotonically increasing. Since \(x_1 > 0\) and \(c > 0\), by induction, \(x_{n+1} > x_n\) for all \(n\), thus the sequence is monotonically increasing.

Step 3: Show the sequence is bounded above

We need to show that there exists an upper bound \(B\) such that \(x_n \leq B\) for all \(n\). In the context of this problem, the solution of the quadratic equation, \(x\), serves as the logical candidate for an upper bound since \(x_{n+1} = \sqrt{c + x_n} \leq \sqrt{c + x} = x\). This holds true because \(x_{n+1}\) only depends on the previous term and \(c\), not on any larger potential value. Thus, the sequence is bounded above.

Step 4: Apply the Monotone Convergence Theorem

Since we have shown that the sequence \(\{x_n\}\) is monotonically increasing and that it is also bounded above, we can invoke the Monotone Convergence Theorem. The theorem states that every bounded, monotonically increasing sequence converges to a limit. This implies that the limit of the sequence \(\{x_n\}\) exists.

Step 5: Find the limit of the sequence

To determine the actual limit of the sequence, \(L\), we know it must satisfy the fixed point condition mentioned in Step 1: \(L = \sqrt{c + L}\), which implies that \(L^2 = c + L\). This is the same as our original quadratic equation: \(L^2 - L - c = 0\). In the context of this problem, the positive root of this equation is relevant since \(x_1 > 0\) and all subsequent \(x_n\) are positive. Therefore, the sequence \(\{x_n\}\) converges to the positive solution of the quadratic equation.

Key Concepts

Monotone Convergence Theorem

In the study of calculus and mathematical analysis, the Monotone Convergence Theorem is a critical concept that offers a gateway to understanding the behavior of certain sequences. Its essence lies in two fundamental properties: monotonicity and boundedness. A sequence is called monotonic if it either never decreases (monotonically increasing) or never increases (monotonically decreasing). Concurrently, a sequence is said to be bounded if there's a limit beyond which the terms of the sequence do not go.

To apply this theorem, you must check two things in the sequence: first, that it is indeed monotonic, and second, that it is bounded above (for increasing sequences) or below (for decreasing sequences). Once these criteria are satisfied, the Monotone Convergence Theorem assures us of the sequence's convergence. Specifically, an increasing sequence that is bounded above will converge to the smallest upper bound, while a decreasing sequence that is bounded below converges to the largest lower bound. This theorem is particularly relevant when analyzing recursively defined sequences, as it provides a formal means to confirm their convergence.

Recursive Sequences in Calculus

Recursive sequences are frequently encountered in the mathematical field of calculus, presenting unique challenges and requiring a distinctive approach to understanding their long-term behavior. A recursive sequence is defined by a starting value and a rule that relates each term to one or more of the preceding terms. Unlike explicit sequences, where a formula exists that directly calculates the nth term, recursive sequences build upon themselves iteratively.

Key to working with these sequences is identifying patterns and behaviors as the terms evolve. The example given in the problem—defining a sequence by the formula \(x_{n+1} = \sqrt{c + x_n}\)—is a classic recursive sequence. By examining the relationship between consecutive terms and utilizing the Monotone Convergence Theorem, one can determine if the sequence will eventually settle into a constant value, known as the limit. Through careful analysis and understanding of the recursive relationship, particularly how it impacts successive terms, mathematicians can unlock the secrets of convergence within these sequences.

Quadratic Equations

Quadratic equations are an integral part of algebra and form the base for various advanced concepts in calculus and other areas of mathematics. They are second-degree polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(a\) is not zero. These equations are notable for their characteristic 'U'-shaped graph, known as a parabola.

The solutions to these equations, often referred to as 'roots', can be real or complex and are represented by the points where the parabola intersects the x-axis. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is widely used to find these roots. In the context of recursive sequences and convergence, the quadratic equation engenders a fixed point that the terms of the sequence may approach—this is the heart of an exercise like the one discussed, linking the realms of algebra and calculus.

Limits of Sequences

The concept of limits is foundational in calculus and serves as a bridge to understanding the behavior of functions and sequences as they approach a particular point or infinity. For sequences specifically, a limit is the value that the terms of the sequence get arbitrarily close to as the index increases without bounds. This is mathematically denoted as \(\lim_{n\to\infty} a_n = L\), expressing that the terms \(a_n\) approach the limit \(L\) as \(n\) becomes very large.

The pursuit of limits often involves proving that a sequence converges, meaning it has a well-defined limit, as opposed to divergence, where no such limit exists. The convergence can sometimes be established by showing that the sequence is both monotonic and bounded as we have seen in the context of the Monotone Convergence Theorem. In the case of recursive sequences, the term immediately before a given term plays a significant role in calculating the next value, and this reliance creates a pattern that, if constrained by the right conditions, leads to a limit, crafting a bridge to the underlying concept of convergence.