Problem 79 - Calculus: Early Transcendentals

Question

Find the limit of the sequence \( \left\\{ \sqrt 2, \sqrt{2\sqrt2}, \sqrt{2\sqrt{2\sqrt2}}, \cdot \cdot \cdot \right\\} \)

Step-by-Step Solution

Answer

The limit of the sequence is 2.

Step 1: Observe the sequence

Let's denote the sequence as \( a_n \). We have \( a_1 = \sqrt{2} \), \( a_2 = \sqrt{2\sqrt{2}} \), \( a_3 = \sqrt{2\sqrt{2\sqrt{2}}} \), and so on. The sequence can be rewritten using a recursive formula: \( a_{n+1} = \sqrt{2a_n} \). This recursive formula will be useful for understanding the behavior of the sequence.

Step 2: Guess the limit

Inspecting the sequence, we notice that as we continue to apply the operation \( a_{n+1} = \sqrt{2a_n} \), the sequence seems to be approaching a fixed value. Let's hypothesize that \( a_n \to L \) as \( n \to \infty \). From the recursive formula, substituting the limit gives \( L = \sqrt{2L} \).

Step 3: Solve the limit equation

From the equation \( L = \sqrt{2L} \), square both sides to remove the square root: \( L^2 = 2L \). Rearranging gives the quadratic equation \( L^2 - 2L = 0 \). Factor the equation to get \( L(L - 2) = 0 \). Thus, \( L = 0 \) or \( L = 2 \).

Step 4: Verify the valid limit

Since \( a_n \) represents positive values from the sequence (being a square root of positive numbers), \( L = 0 \) can be disregarded as a possibility for the limit. The sequence values and their progression suggest \( L = 2 \) is the only feasible limit. Thus, the limit of the sequence is 2 for \( n \to \infty \).

Key Concepts

recursive sequences

A recursive sequence is defined using a rule or formula that relates each term to one or more of its preceding terms. In this exercise, the sequence is defined using the recursive formula \( a_{n+1} = \sqrt{2a_n} \). This means each term is determined based on the value of the previous term. Recursive sequences are common in mathematics as they provide a systematic way to generate sequences based on initial known values, often referred to as base cases.

• To understand recursive sequences, start by identifying the initial term(s), known as base cases.
• Apply the recursive rule to generate further terms in the sequence.
• Recursive sequences help explore interesting behaviors such as convergence or oscillation.

In our specific problem, we observed that starting with \( a_1 = \sqrt{2} \), each subsequent term involves taking the square root of double the previous term, gradually approaching specific behavior that leads us to explore the concept of limits.

limits of sequences

The limit of a sequence addresses what the sequence approaches as the index goes to infinity. Mathematically, for a sequence \( \{a_n\} \), we say the limit is \( L \) if \( a_n \to L \) as \( n \to \infty \). Finding the limit involves looking for a value that the sequence terms increasingly approach, but never necessarily reach precisely.

• In the exercise provided, we hypothesize that \( a_n \to L \) with the recursive relationship \( L = \sqrt{2L} \).
• Solve \( L = \sqrt{2L} \) by first squaring both sides to eliminate the square root, forming a solvable equation.

In this case, solving such an equation helps determine potential limit values. By further logical reasoning, we refine the possibilities to find the most plausible limit. In the example, we consider only positive values and establish the limit \( L = 2 \) for the sequence \( a_n \).

convergence of sequences

Convergence of a sequence refers to the property of the sequence approaching a fixed value, or limit, as the index increases indefinitely. When a sequence converges, the terms become arbitrarily close to a specific value. In our exercise, determining convergence involved examining the recursive formula and solving the associated limit equation.

• Inspect sequence behavior through the recursive formula \( a_{n+1} = \sqrt{2a_n} \), noticing how the terms stabilize.
• Evaluate the outcomes of the limit equation \( L^2 - 2L = 0 \) to identify plausible limits.

To determine convergence, we must ensure all potential limits satisfy the sequence's requirements. Positive terms in our sequence allow us to disregard \( L = 0 \) as a viable option, affirming the sequence converges to the value \( 2 \). Understanding convergence helps in proving robustness in mathematical modeling and predictions based on sequential behaviors.