Question

Determine whether the sequence defined as follows has a limit. If it does, find the limit. \(a_{1}=\sqrt{2}, a_{2}=\sqrt{2 \sqrt{2}}, a_{3}=\sqrt{2 \sqrt{2 \sqrt{2}}}\) et

Short answer

The sequence has a limit, and the limit is 2.

Step-by-step solution

Identify the Sequence Pattern

The given sequence is \(a_1 = \sqrt{2} \), \( a_2 = \sqrt{2 \sqrt{2}} \), \( a_3 = \sqrt{2 \sqrt{2 \sqrt{2}}} \), and so on. Notice that each term can be expressed as \( a_n = \sqrt{2 a_{n-1}} \).

Identify the Limit Formation

To solve whether the sequence has a limit, assume \( a_n \) approaches a limit \( L \). Then, as \( n \to \infty \), \(a_{n+1} = \sqrt{2 a_n} \) should also approach \( L \). Therefore, the equation \( L = \sqrt{2L} \) holds.

Solve the Equation for the Limit

From the equation \( L = \sqrt{2L} \), square both sides to get \( L^2 = 2L \). Rearrange the equation to \( L^2 - 2L = 0 \), which can be factored to \( L(L - 2) = 0 \).

Determine the Possible Limits

The above equation gives two potential limits: \( L = 0 \) or \( L = 2 \).

Consider the Sequence Values

Considering \(a_1 = \sqrt{2} > 0 \) and since each subsequent term is formed by taking the square root of a positive number, all terms \(a_n\) are positive. Therefore, \( L eq 0 \).

Conclude the Limit

Given that \( L eq 0 \) and the potential values were \( L = 0 \) or \( L = 2 \), the limit of the sequence must be \( L = 2 \).

Key concepts

Convergence

Convergence is a fundamental concept in the study of sequences and series. When we say a sequence "converges," we mean that as you go further into the sequence, the terms get closer and closer to a specific number. This specific number is called the "limit" of the sequence.
For example, in the sequence given in the exercise, each term is defined based on the previous one, structured in such a way that it eventually stabilizes to a single value. Understanding convergence helps us determine whether it's meaningful to discuss the endpoint (or limit) of an infinite sequence.
To check if a sequence converges, we can often look for a recursive formula or relationship among terms that simplifies over time. If the terms of a sequence approach a particular value, leading them to circle closer with increasingly smaller deviations, we say the sequence converges, and that value is its limit.

• Convergence translates intuitive reasoning into a formal prediction about sequences.
• It provides a robust method to understand infinite processes by focusing on their behavior at infinity.

Recognizing whether a sequence converges is crucial for exploring more complex mathematical entities, such as series and calculus operations.

Recursive Sequences

Recursive sequences are those in which each term is defined based on one or more previous terms. This definition allows us to build a sequence step-by-step by applying a fixed rule repeatedly. In the context of the exercise, the recursive relation is identified as each term in the sequence is expressed with a dependence on the previous term: \( a_n = \sqrt{2 a_{n-1}} \).
Understanding recursive sequences is essentially predicting the long-term behavior of a process that builds on itself. Many natural processes and mathematical models are expressed in recursive forms, emphasizing their importance.

• The rule itself is usually simple, but the resulting sequence may exhibit complex behavior.
• Recursive formulas allow us to translate real-world problems into mathematical formats that can be manipulated and solved.

Recursive sequences often require clever algebraic manipulation to determine properties like convergence or limits, as shown when solving for the sequence's limit in the exercise. Picking apart these relationships is a cornerstone of understanding complex systems in calculus and beyond.

Limits in Calculus

Limits are at the heart of calculus and underline a vast array of concepts within the discipline. In sequences, finding the limit involves determining the value a sequence approaches as the terms extend towards infinity.
For instance, with our sequence, we established that as n becomes very large, the terms of the sequence approach the value 2. We did this by analyzing the relationship \( a_{n+1} = \sqrt{2 a_n} \) and addressing it through the equation \( L = \sqrt{2L} \). Solving this gave us potential limits which further analysis showed to be 2.

• Limits help us understand the behavior of sequences and functions that do not stop growing or changing.
• They form the base for defining other aspects of calculus like continuity, derivatives, and integrals.
• The precision of limits allows us to deal with infinite processes by focusing on their results.

In calculus, mastering limits means being able to tackle diverse problems, exploring functions' behaviors, and predicting movement and growth within a mathematical framework.