Problem 109 - Calculus for Biology and Medicine

Question

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed ooints of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which ixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{4}{a_{n}}\right), a_{0}=1 $$

Step-by-Step Solution

Answer

The sequence converges to the fixed point 2.

Step 1: Understand the Fixed Point Equation

To determine the fixed points of the sequence, set the equation for the sequence equal to itself: \[ a = \frac{1}{2}\left(a + \frac{4}{a}\right) \]Solving this will allow us to find the values of \(a\) that don't change across iterations.

Step 2: Solve the Fixed Point Equation

Multiply both sides by 2 to eliminate the fraction:\[ 2a = a + \frac{4}{a} \]Subtract \(a\) from both sides:\[ a = \frac{4}{a} \]Multiply both sides by \(a\) to clear the fraction:\[ a^2 = 4 \]Take the square root of both sides to find the potential fixed points:\[ a = \pm 2 \]

Step 3: Analyze the Sequence Behavior

Investigate the behavior of the sequence starting from the initial condition \(a_0 = 1\). We compute the next few terms:- \( a_1 = \frac{1}{2}(1 + \frac{4}{1}) = \frac{5}{2} = 2.5 \)- \( a_2 = \frac{1}{2}(2.5 + \frac{4}{2.5}) \approx 2.05 \)This suggests the sequence is converging to 2.

Step 4: Conclude on the Limiting Value

Since the sequence appears to be converging to 2, and we found that \(2\) is one of the fixed points, we conclude that the sequence starting from the initial condition \(a_0 = 1\) converges to 2.

Key Concepts

Fixed Points

Fixed points in a sequence are values that remain unchanged with each iteration of the sequence's function. In other words, if a sequence takes in a value and returns the same value, then that value is a fixed point.

Let's consider the expression given: \( a_{n+1} = \frac{1}{2}\left(a_{n} + \frac{4}{a_{n}}\right) \). A fixed point \( a \) satisfies \( a = \frac{1}{2}\left(a + \frac{4}{a}\right) \). To find the fixed points, we solve this equation:

• Multiply both sides by 2 to clear the fraction: \( 2a = a + \frac{4}{a} \).
• Subtract \( a \) from both sides: \( a = \frac{4}{a} \).
• Multiply by \( a \) to eliminate the fraction: \( a^2 = 4 \).
• Finally, take the square root of both sides: \( a = \pm 2 \).

We now know that the fixed points are 2 and -2. These fixed points tell us about possible stable states of the sequence. Analyzing the behavior of the sequence will show which of these fixed points the sequence approaches.

Limits

The concept of limits helps us understand the behavior of a function or sequence as it approaches a particular point. When we say a sequence \( \{a_n\} \) has a limit, it means that as \( n \) becomes very large, the terms \( a_n \) get closer and closer to some fixed number.

In the given problem, the sequence is \( a_{n+1} = \frac{1}{2}\left(a_n + \frac{4}{a_n}\right) \) with the initial condition \( a_0 = 1 \). As we compute the next terms starting from \( a_0 \):

• \( a_1 = \frac{1}{2}(1 + \frac{4}{1}) = 2.5 \)
• \( a_2 = \frac{1}{2}(2.5 + \frac{4}{2.5}) \approx 2.05 \)
• Continuing this process, the numbers appear to approach 2.

The sequence seems to be stabilizing around 2, suggesting that the limit of the sequence is 2. This is consistent with our finding that 2 is a fixed point, strengthening our understanding of its convergence.

Sequence Convergence

Sequence convergence refers to the process by which the terms of a sequence approach a specific value.
In our context, when we say that \( \{a_n\} \) converges, it means each successive term gets closer to a certain number, known as the limit.

Starting from \( a_0 = 1 \), the sequence shows:

• \( a_1 = 2.5 \)
• \( a_2 \approx 2.05 \)
• ...and it continues closer toward 2.

Convergence is a crucial concept as it assures us the sequence isn't just erratically fluctuating but instead heading toward a specific target value.

With our sequence, each step seems to reduce the difference from 2, illustrating clear convergence toward it. Given the fixed points were found to be \( \pm 2 \), starting from \( a_0 = 1 \), the positive direction was chosen, making 2 the natural choice as the limit for our given sequence.