Problem 84 - Calculus: Early Transcendentals

Question

(a) Let \( a_1 =a, a_2 = f(a), a_3 = f(a_2) = f( f(a)), . . . , a_{n + 1} = f(a_n), \) where \( f \) is a continuous function. If \( lim_{n \to\infty} a_n = L, \) show that \( f(L) = L. \) (b) Illustrate part (a) by taking \( f(x) = \cos x, a = 1, \) and estimating the value of \( L \) to five decimal places.

Step-by-Step Solution

Answer

(a) \( f(L) = L \) by continuity. (b) \( L \approx 0.73908 \).

Step 1: Understand the Limit Definition

We are given the sequence \( a_1 = a, a_2 = f(a), a_3 = f(a_2), \ldots, a_{n+1} = f(a_n) \). We know the limit of this sequence, \( \lim_{n \to\infty} a_n = L \), exists. This means as \( n \) increases, the terms of the sequence become arbitrarily close to \( L \).

Step 2: Apply Function Continuity

Given that \( f \) is continuous, for sequences \( a_n \) converging to \( L \), we have \( \lim_{n \to \infty} f(a_n) = f(\lim_{n \to \infty} a_n) = f(L) \).

Step 3: Show \( f(L) = L \)

Since \( a_{n+1} = f(a_n) \) and \( \lim_{n \to \infty} a_n = L \), we also have \( \lim_{n \to \infty} a_{n+1} = L \). By continuity, \( \lim_{n \to \infty} f(a_n) = f(L) = L \). Thus, \( f(L) = L \).

Step 4: Illustrate with Example Function

For part (b), set \( f(x) = \cos x \) and \( a = 1 \). Compute the sequence: \( a_1 = 1, a_2 = \cos 1, a_3 = \cos(a_2), \) etc. Using a calculator, iterate this process a few times to estimate \( L \).

Step 5: Calculate Numerical Estimate for \( L \)

Iterate the process \( a_{n+1} = \cos(a_n) \) until \( |a_{n+1} - a_n| < 0.00001 \):1. \( a_1 = 1 \)2. \( a_2 = \cos 1 \approx 0.5403 \)3. \( a_3 = \cos(0.5403) \approx 0.8576 \)4. Continue iterating until the difference is negligible. The sequence converges to approximately 0.73908.

Key Concepts

Continuous Function

A continuous function is a fundamental concept in calculus and analysis. At its core, continuity means that a function's value changes smoothly without any abrupt jumps or breaks. To understand this, think of drawing a curve on a graph without lifting your pen: this curve represents a continuous function.

For a function \( f(x) \) to be continuous at a point \( c \), the following must hold true:

• The function \( f(x) \) is defined at \( c \).
• There exists a limit \( \lim_{x \to c} f(x) \) and the limit exists.
• The value of the function at that point is equal to the limit: \( f(c) = \lim_{x \to c} f(x) \).

These conditions ensure that as \( x \) approaches the point \( c \), \( f(x) \) approaches \( f(c) \) as well.

For example, in the given exercise, the continuity of the function \( f \) ensures that you can apply limits effectively, guaranteeing that the limit of a sequence applies directly to the function value at that limit.

Sequence Convergence

Sequence convergence is an essential concept when dealing with sequences in mathematics. A sequence is simply a string of numbers arranged in a specific order. A sequence converges if it approaches some fixed value, known as the limit, as the sequence progresses (i.e., as the number of terms increases).

To say that a sequence \( \{a_n\} \) converges to a limit \( L \), means:

• For any small positive number \( \epsilon \) (no matter how tiny), there exists a term in the sequence such that from this point onward, all subsequent terms \( a_n \) are within an \( \epsilon \) distance from \( L \).
• This translates mathematically to: given \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( n \geq N \), \(|a_n - L| < \epsilon \).

In our exercise, the sequence \( a_n \) is constructed through iterations of the function \( f \), starting from a given point \( a \). The sequence is shown to converge to a limit \( L \) by the properties of the continuous function \( f \). Through iterative evaluations, each term gets closer to the limit \( L \).

Limit of a Sequence

The limit of a sequence is a concept that identifies where a sequence is headed as the number of terms becomes exceedingly large. In mathematical terms, if a sequence \( \{a_n\} \) has a limit \( L \), it means that the values of \( a_n \) get arbitrarily close to \( L \) as \( n \) becomes larger and larger.

The formal mathematical definition of the limit of a sequence is:

• For any given \( \epsilon > 0 \), no matter how small, there exists a natural number \( N \) such that for all \( n \geq N \), the absolute difference \(|a_n - L| < \epsilon \).

In our particular problem, the sequence \( a_n \), which emerges through repeated application of the function \( f \), approaches a number \( L \). The continuous nature of \( f \) ensures that at this limit, the function and the sequence converge to the same value. Essentially, as \( n \) grows indefinitely, \( a_n \) approaches \( L \), and thus \( f(L) = L \), which affirms the fixed point theorem referenced in the solution.