Problem 41 - Essential Calculus Early Transcendentals

Question

Find the limit of the sequence $$\\{\sqrt{2}, \sqrt{2 \sqrt{2}}, \sqrt{2 \sqrt{2 \sqrt{2}}}, \ldots\\}$$

Step-by-Step Solution

Answer

The limit of the sequence is 2.

Step 1: Recognize the Sequence Pattern

The sequence is given as \(a_1 = \sqrt{2}\), \(a_2 = \sqrt{2\sqrt{2}}\), \(a_3 = \sqrt{2\sqrt{2\sqrt{2}}}\), and so forth. We recognize the recursive pattern where \(a_{n+1} = \sqrt{2a_n}\).

Step 2: Formulate the General Term

Using the recursive relation, each term can be expressed as: \(a_{n+1} = \sqrt{2a_n}\). We now aim to find a limit, \(L\), as \(n\to\infty\).

Step 3: Assume the Limit Exists

Assume the limit \(L\) exists, meaning \(\lim_{n\to\infty} a_n = L\). If the limit exists, substituting into the recursive formula gives us \(L = \sqrt{2L}\).

Step 4: Solve the Limit Equation

We solve the equation \(L = \sqrt{2L}\), by squaring both sides to get: \[ L^2 = 2L \]Rearrange this to \[ L^2 - 2L = 0 \]Factor this equation: \[ L(L - 2) = 0 \]Thus, \(L = 0\) or \(L = 2\).

Step 5: Determine the Valid Limit

Initially, \(a_1 = \sqrt{2} > 0\). Therefore, all terms \(a_n > 0\) for \(n\geq 1\). Thus, \(L = 0\) is not possible as it contradicts the positivity of each term. Therefore, the limit of the sequence is \(L = 2\).

Key Concepts

Recursive Sequences

In mathematics, recursive sequences are sequences where each term is defined as a function of its preceding term(s). They establish a pattern that can be followed indefinitely. Understanding recursive sequences can be quite beneficial as they simplify complex patterns and make predictions easier by only needing an initial value and a rule for the sequence to progress.

For the sequence presented, start with the initial term given as \(a_1 = \sqrt{2}\). Following this, each subsequent term is dependent on the one before it and is defined by the recursive relationship \(a_{n+1} = \sqrt{2a_n}\). This specifically defines how the sequence progresses.

• The sequence begins with a known value.
• Each next term uses the preceding term(s).
• This type of sequence can convey growth or decay, depending on the formula used.

Understanding recursive relations is significant in determining the dynamic behavior of sequences as it clarifies how initial conditions influence long-term trends.

Limit of a Sequence

The concept of the limit of a sequence is fundamental in mathematical analysis and calculus. It refers to the value that the terms of a sequence tend to as the sequence goes on indefinitely, that is, as \(n\) approaches infinity.

For the sequence given in the exercise, we assume a limit \(L\) exists for \(\lim_{n\to\infty} a_n\). Assuming the sequence converges (which happens if a sequence settles on a specific number), we employ the recursive formula here to find the limit by substituting it, leading to \(L = \sqrt{2L}\).

Solving the limit equation involves:

• Squaring both sides: transforms \(L = \sqrt{2L}\) into \(L^2 = 2L\).
• Solving the quadratic equation \(L^2 - 2L = 0\).
• Factoring the equation, yielding \(L(L - 2) = 0\).

The valid solution \(L = 2\) is obtained considering that all sequence terms are positive, aligning with the rule and structure of the sequence given.

Square Root Sequence

A square root sequence, as in this exercise, involves each term being derived from the square root of the previous term or terms, often coupled with additional operations.

The sequence in question is specifically a square root sequence where each term \(a_{n+1}\) is calculated as \(a_{n+1} = \sqrt{2a_n}\). It characterizes sequences designing growth that slows over time. Using roots tends to smooth the progression, dampening large values back down to scale.

Key properties of square root sequences:

• They usually converge to a limit if their structure ensures bounded and monotonic behavior.
• Often, they approach a fixed point or limit as demonstrated with the limit \(L = 2\) in previous sections.
• It's common in square root sequences that the early terms may vary greatly, but they traditionally either stabilize or oscillate between values.

By examining such sequences, one appreciates how initial conditions and recursive operations influence the behavior and outcome of the entire sequence.