Problem 42 - Essential Calculus Early Transcendentals

Question

A sequence \(\left\\{a_{n}\right\\}\) is given by \(a_{1}=\sqrt{2}, a_{n+1}=\sqrt{2+a_{n}}\) (a) By induction or otherwise, show that \(\left\\{a_{n}\right\\}\) is increasing and bounded above by \(3 .\) Apply the Monotonic Sequence Theorem to show that lim \(_{n \rightarrow \infty} a_{n}\) exists. (b) Find \(\lim _{n \rightarrow \infty} a_{n} .\)

Step-by-Step Solution

Answer

The sequence is increasing, bounded by 3, and converges to 2.

Step 1: Understanding the sequence definition

The given sequence is \(a_{1} = \sqrt{2}\) and \(a_{n+1} = \sqrt{2 + a_n}\). We aim to prove that this sequence is increasing and bounded above by 3, proving that it converges.

Step 1: Prove the sequence is increasing

To show that \(\{a_n\}\) is increasing, we need to prove that \(a_{n+1} \geq a_n\). Starting from the recurrence relation, verify:\[ a_{n+1} = \sqrt{2 + a_n} \geq a_n \]Squaring both sides:\[ 2 + a_n \geq a_n^2 \]Rearrange to:\[ a_n^2 - a_n - 2 \leq 0 \]Solve the quadratic \(a_n^2 - a_n - 2 = 0\) to find roots:\[ a_n = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2} \Rightarrow a_n = 2 \text{ or } -1\]Since \(a_n > 0\), use root 2 and verify \(a_1 = \sqrt{2} \approx 1.41 < 2\), hence the inequality holds initially. Therefore, \(a_n\) is increasing for all \(n\).

Step 2: Prove the sequence is bounded above by 3

Prove \(a_n \leq 3\) for all \(n\). Assume \(a_n \leq 3\), Then \(a_{n+1} = \sqrt{2 + a_n} \leq \sqrt{2+3} = \sqrt{5} \approx 2.24\).Hence, if the sequence starts below 3, every subsequent term is less than 3.Since \(a_1 = \sqrt{2} \approx 1.41\) and \(a_2 = \sqrt{2 + \sqrt{2}} \approx 1.85\), both are clearly less than 3.This proves by induction that \( a_n \leq 3 \) for all \(n\).

Step 3: Apply the Monotonic Sequence Theorem

Since the sequence \( \{a_n\} \) is shown to be increasing and bounded by 3, by the Monotonic Sequence Theorem, \( \lim_{n \to \infty} a_n \) exists.

Step 4: Find the limit of the sequence

Assume the limit exists as \(L\). Taking limits on both sides of the recurrence equation:\[ L = \sqrt{2 + L} \]Square both sides:\[ L^2 = 2 + L \]Rearrange to solve the quadratic:\[ L^2 - L - 2 = 0 \] Using the quadratic formula:\[ L = \frac{1 \pm \sqrt{1+8}}{2} = \frac{1 \pm 3}{2} \Rightarrow L = 2 \text{ or } -1 \]Since the sequence is positive, the limit is \(L = 2\).Thus, \( \lim_{n \to \infty} a_n = 2 \).

Key Concepts

Increasing Sequence

An increasing sequence is a sequence of numbers where each term is greater than or equal to the term before it. To determine if a sequence is increasing, we check that for every pair of consecutive terms, the latter is greater than or equal to the former. This property signifies consistency in growth or constancy in the values of the sequence over time.

• For the given sequence \(a_{n+1} = \sqrt{2 + a_n}\), we need to show that \(a_{n+1} \geq a_n\).
• By calculating for specific terms and through algebraic manipulation, it is shown that \(a_n\) grows over time.

In our example, since \(a_1 = \sqrt{2}\) and the structure of the sequence guides each subsequent term to come closer to the regularity defined by this inequality, it initiates an increasing pattern throughout the sequence. Understanding that a sequence grows or maintains its value helps us in predicting the direction of the sequence in the long run. This behavior is crucial in establishing the sequence's convergence later on.

Bounded Sequence

A bounded sequence is one where all terms fall within a specific fixed range. For a sequence to be bounded above means that no term in the sequence exceeds a certain value. This feature is vital as it restricts the possible outcomes and establishes stability for the sequence.

• In the provided sequence, we sought to prove that \(a_n \leq 3\) for all \(n\).
• This means every term \(a_n\) will never surpass 3, ensuring no part of the sequence shoots indefinitely high.
• With the starting term \(a_1 = \sqrt{2}\) and subsequent calculations showing progressions that stay below 3, it confirms this bounded behavior.

Bounding provides a consistent frame within which the sequence operates, ensuring predictability of terms. When determining the convergence of sequences, identifying if it's bounded is just as crucial as understanding its direction (increasing or decreasing). This prevents the terms from escaping an infinite range, a prerequisite for evaluating the sequence’s convergence.

Limit of a Sequence

The limit of a sequence refers to the value that the terms of a sequence get closer to as the sequence progresses indefinitely. When we know a sequence is bounded and increasing, these properties significantly guide us to predict and establish its limit.

• In the example provided, we used the equation \(L = \sqrt{2 + L}\) to find the limit \(L\) of the sequence.
• Upon solving the rearranged equation \(L^2 - L - 2 = 0\), the roots provide potential limits.
• Considering only positive values due to sequence properties, \(L = 2\) emerges as the valid limit.

The idea of the limit captures the eventual approach and settlement of the sequence’s terms, reflecting stable behavior as the sequence extends. Determining a sequence’s limit is instrumental in understanding long-term trends and predictions of sequential processes, aligning with the Monotonic Sequence Theorem’s implications.