Problem 113 - Calculus for Biology and Medicine

Question

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\sqrt{2 a_{n}}, a_{0}=1 $$

Step-by-Step Solution

Answer

The fixed points are 0 and 2. The sequence converges to the fixed point 2.

Step 1: Define a Fixed Point

A fixed point of the sequence \(\{a_n\}\) is a value \(x\) such that \(a_{n+1} = a_n = x\). This means when the sequence reaches this point, it will remain constant. Therefore, for our recurrence relation, we need to solve \(x = \sqrt{2x}\).

Step 2: Solve for Fixed Points

To solve \(x = \sqrt{2x}\), square both sides to eliminate the square root: \(x^2 = 2x\). Rearrange the equation to \(x^2 - 2x = 0\). This can be factored as \(x(x - 2) = 0\). Thus, \(x = 0\) and \(x = 2\) are the fixed points.

Step 3: Analyze Behavior of the Sequence

Starting from \(a_0 = 1\), compute a few terms to see the sequence progression: - \(a_1 = \sqrt{2a_0} = \sqrt{2}\)- \(a_2 = \sqrt{2a_1} = \sqrt{2\sqrt{2}}\)- Continue calculating to observe the sequence approaches a limit. The sequence is strictly increasing starting from \(a_0 = 1\).

Step 4: Consider Limits and Reasoning

As \(a_n\) is increasing and bounded above by 2 (since \(a_{n+1} = \sqrt{2a_n} \leq 2\) due to the square root), it converges to a limit. Between the two fixed points, 0 and 2, the sequence is clearly approaching 2 as \(n o \infty\) given it started from 1 and is increasing.

Key Concepts

Limit of a Sequence

In mathematics, a sequence is a list of numbers following a particular order or pattern. When we talk about the limit of a sequence, we are interested in the behavior of the sequence as its index approaches infinity.

The limit of a sequence \(a_n\) is the value that the terms of the sequence "tend" toward as \(n\) grows larger. If a sequence reaches such a stable point, it is said to converge.

• Consider the sequence given by \(a_{n+1} = \sqrt{2a_n}\).
• If as \(n \to \infty\), the terms stop changing and approach a particular number, that number is the limit.

For example, in the given exercise, the sequence begins with \(a_0 = 1\). The task is finding what value the sequence approaches as \(n\) increases. After analyzing the progression of terms, it psychologically draws closer to the limit of 2. Understanding how to identify this limit involves recognizing patterns or using mathematics to verify fixed points, as done in the solution.

Recursive Sequence

A recursive sequence is a sequence in which each term is defined as a function of its preceding terms. The defining relation shows how each term relates to its predecessor, often improving our understanding of the sequence's behavior.

In the exercise, the recursive formula is \(a_{n+1} = \sqrt{2a_n}\).

• This suggests that any term is the square root of twice the previous term.
• Starting with an initial condition, \(a_0 = 1\), the sequence generates its succeeding terms based on this recursive rule.

Recursive sequences are significant in various applications, from algorithm design in computer science to solving real-world problems in economics and biology. Recognizing the recursive nature helps in deducing general properties and identifying long-term behavior as \(n\) approaches infinity.

Convergence of Sequences

Convergence is a core concept when dealing with sequences since determining whether a sequence converges tells a great deal about its behavior at infinity.

A sequence \(a_n\) converges if it approaches a specific number as \(n\) becomes very large, meaning subsequent terms get arbitrarily close to this number.

• For convergence to occur, it is essential the sequence does not oscillate indefinitely or diverge to infinity.
• In our exercise, the sequence \(a_{n+1} = \sqrt{2a_n}\) is bounded above by 2. Therefore, while starting at 1, uniformly increasing ensures it converges to 2, one of its fixed points.

Understanding convergence allows mathematicians to "predict" the long-term behavior of sequences beyond just calculating numerous terms. When a sequence is proven to converge, like the one in our problem, it confirms the existence of a limit, simplifying further analysis or application.