Problem 7 - A Textbook of Differential Calculus

Question

If \(x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}\) and \(\left|x_{0}\right|<1, n \geq 0\), then \(\lim _{n \rightarrow \infty}\left(\frac{\sqrt{1-x_{0}^{2}}}{x_{1} x_{2} x_{3} \ldots x_{n+1}}\right)\) is equal to (a) \(-1\) (b) 1 (c) \(\cos ^{-1}\left(x_{0}+1\right)\) (d) \(\cos ^{-1}\left(x_{0}\right)\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Initial Sequence Definition

The given sequence is defined as \( x_{n+1} = \sqrt{\frac{1 + x_n}{2}} \). This is a function of the previous term, requiring us to find a pattern or behavior as \( n \) approaches infinity.

Step 2: Analyzing the Convergence

Since \( |x_0| < 1 \), it implies the sequence \( x_n \) remains bounded and strictly less than 1 for all \( n \). This feature often points towards the convergence of the sequence towards some limit.

Step 3: Recognizing the Sequence Limit

The structure \( x_{n+1} = \sqrt{\frac{1 + x_n}{2}} \) resembles the recursive generation of values converging to \( \cos(\theta) \), particularly in trigonometric series, which often stabilize towards fixed values.

Step 4: Relating to Cosine Function

Assume that \( x_n = \cos(\theta_n) \). In trigonometric identities, the relation \( \cos(2\theta) = 2\cos^2(\theta) - 1 \) and its manipulation to form sequences that simplify to cosines is useful here. The limit of interest connects with the cosine inverse.

Step 5: Infinity Limit and Assumptions

Derive that as \( n \rightarrow \infty \), the terms stabilize such that \( x_n \rightarrow \cos(\theta) \). The limit \( \lim _{n \rightarrow \infty} \left(\frac{\sqrt{1-x_{0}^{2}}}{x_{1} x_{2} x_{3} \ldots x_{n+1}}\right) \) can then be simplified using the properties of the cosine and its inverse due to the recursive structure.

Step 6: Final Calculation and Answer Explanation

Thus, by applying the properties of convergence in such sequences and known behavior: \[ \lim _{n \rightarrow \infty}\left(\frac{\sqrt{1-x_{0}^{2}}}{x_{1} x_{2} x_{3} \ldots x_{n+1}}\right) = 1 \]. The result is based on the trigonometric relationship derived from initial assumptions and transformations.

Key Concepts

Convergence of a Sequence

In mathematics, the convergence of a sequence refers to its behavior as the index approaches infinity. A sequence is said to converge if it tends toward a particular value. For our sequence given by \(x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}\), understanding convergence is crucial. Since the initial condition \(|x_0| < 1\) implies that each term stays within certain bounds, this is an indication of the sequence's tendency to converge. When dealing with recursive sequences like this one, convergence is often assumed when the terms do not appear to 'escape' to infinity or oscillate indefinitely. In this exercise, the sequence \(x_n\) continues to get closer to a fixed value as \(n\) grows larger. This behavior signifies that a limit exists, which further enables us to predict values and resolve terms to their eventual behavior.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved variables where they are defined. They are incredibly useful in simplifying expressions involving trigonometric functions. In our sequence, the relation \(x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}\) hints at a trigonometric underpinning.By applying identities such as \(\cos(2\theta) = 2\cos^2(\theta) - 1\), we can transform our sequence into a form that is easier to analyze and understand. The recognition that the sequence is behaving akin to a cosine function is a powerful insight that leads us to find the limit of the values in terms of known trigonometric properties.

Cosine Function

The cosine function is a fundamental trigonometric function often denoted as \(\cos(\theta)\). In the context of our sequence, we assume \(x_n=\cos(\theta_n)\), which helps us relate the sequence to familiar trigonometric concepts.Cosine functions have periodic behavior and a defined amplitude, which means that sequences resembling or derived from cosine functions tend to exhibit stable behavior. By employing the relation \(x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}\) in terms of cosine functions, we understand that this recursive relationship might stabilize to \(\cos(\theta)\), where \(\theta\) is some constant angle that the sequence essentially approaches as \(n\) tends to infinity.

Recursive Sequences

Recursive sequences are defined with terms that rely on prior terms for their calculation. They can represent a wide range of mathematical constructs and behaviors. Our sequence, defined as \(x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}\), is a simple yet intriguing example of a recursive sequence.Such sequences often require an examination of patterns to predict future behavior and deduce their limits. The recursive nature of \(x_{n+1}\) means each term in the sequence depends on the previous term, leading to a chain reaction that progresses toward a specific value or behavior.Analyzing recursive sequences involves understanding how small changes in initial values can impact the entirety of the sequence, a concept known as sensitivity. This requires careful mathematical manipulation and sometimes, as seen in this exercise, leveraging trigonometric properties to unearth the sequence's intrinsic trends.