Problem 82 - Calculus: Early Transcendentals

Question

Show that the sequence defined by \( a_1 = 2 \) \( a_{n + 1} = \frac {1}{3 - a_n} \) satisfies \( 0 < a_n \le 2 \) and is decreasing. Deduce that the sequence is convergent and find its limit.

Step-by-Step Solution

Answer

The sequence converges to \( \frac{3 - \sqrt{5}}{2} \).

Step 1: Show Base Case for Sequence

For the given sequence, we know that \( a_1 = 2 \). This clearly satisfies the condition \( 0 < a_1 \leq 2 \).

Step 2: Show Inductive Step for Sequence Bound

Assume for some \( n \), \( 0 < a_n \leq 2 \) is true. Then \( a_{n+1} = \frac{1}{3 - a_n} \). Since \( 3 - a_n \geq 1 \), it follows that \( a_{n+1} = \frac{1}{3 - a_n} \leq \frac{1}{1} = 1 \). Thus, \( 0 < a_{n+1} \leq 1 \), showing the inductive step is true.

Step 3: Show the Sequence is Decreasing

Consider \( a_{n+1} = \frac{1}{3 - a_n} \) and \( a_n > a_{n+1} \). By the assumption \( a_n \leq 2 \), we see \( 3 - a_n \geq 1 \) and hence \( a_{n+1} = \frac{1}{3 - a_n} \) becomes smaller as \( a_n \) increases. This implies that the sequence is decreasing.

Step 4: Deduction of Convergence

Since the sequence is bounded below by 0 and is decreasing, it converges by the monotone convergence theorem.

Step 5: Find the Limit

Assume the sequence converges to a limit \( L \). Then \( L = \frac{1}{3 - L} \). Solving this equation: \[ L(3 - L) = 1 \implies 3L - L^2 = 1 \implies L^2 - 3L + 1 = 0 \]The quadratic formula gives: \[ L = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2} \]Since \( 0 < L \leq 2 \), the solution is \( L = \frac{3 - \sqrt{5}}{2} \).

Key Concepts

Monotone Convergence Theorem

The Monotone Convergence Theorem is a fundamental tool in real analysis used to establish the convergence of sequences. This theorem applies to sequences that are both monotonic and bounded. "Monotonic" means that the sequence is either non-increasing or non-decreasing. When coupled with the property of being "bounded," the theorem ensures convergence, making it an essential concept for analyzing sequences.

In our sequence example, we know it is decreasing, meaning each term is less than or equal to the preceding one. The sequence is also bounded. Given these two properties, the Monotone Convergence Theorem can be applied, confirming that the sequence converges to a real number. This conclusion is crucial when solving sequence problems, as it helps simplify complex proofs of convergence. Understanding this theorem aids in tackling similar mathematical scenarios with assurance and clarity.

Bounded Sequences

A sequence is termed "bounded" if there exists a number that acts as an upper limit and another as a lower limit for all terms in the sequence.

For the sequence \( a_n \), the given task is to confirm that each term is greater than 0 and no greater than 2. By analyzing the formula \( a_{n+1} = \frac{1}{3 - a_n} \), we showed that \( 0 < a_n \leq 2 \).

This is achieved by demonstrating that even as the sequence progresses, each term remains within this range. Bounded sequences are easier to handle as they assure that the values stay within predictable limits. This property is fundamental when using the Monotone Convergence Theorem as it secures convergence.

Inductive Proof

Inductive proof is a common method of demonstrating the validity of statements for whole numbers. This mathematical technique involves two main steps:

• The base case establishes the result for the initial value—here, \( a_1 = 2 \).
• The inductive step assumes the result is true for some \( n \), and then demonstrates it holds for \( n+1 \) too.

In our example, starting with the base case \( a_1 = 2 \) shows that \( 0 < a_1 \leq 2 \). At the inductive step, assuming \( 0 < a_n \leq 2 \), we deduced that \( 0 < a_{n+1} \leq 2 \) using the sequence formula. This logical reasoning completes the proof. Inductive proofs are powerful, offering a framework for confirming that conditions hold across an infinite sequence. It makes verifying mathematical claims robust and logical.

Limit of a Sequence

The limit of a sequence refers to a value that the terms in the sequence approach as the number of terms goes to infinity. It is the centerpiece of sequence analysis. Finding a sequence's limit involves a combination of algebraic manipulation and theoretical understanding.

For our decreasing and bounded sequence, we assume it converges to a limit \( L \). We use the defining relation \( L = \frac{1}{3-L} \) and solve the equation to derive possible values for \( L \). This results in a quadratic equation: \[ L^2 - 3L + 1 = 0 \]. Solving this gives us two potential solutions. By considering the bounds \( 0 < L \leq 2 \), we identify the valid limit as \( L = \frac{3 - \sqrt{5}}{2} \).

Understanding how limits work is crucial for comprehending convergence and calculating the eventual outcome of sequences. It eases the exploration of how sequences behave—to predict and effectively utilize the mathematical findings.