Suppose That A Sequence Is Defined As Follows

Let's explore sequences defined by a specific rule or pattern. Understanding these sequences requires a grasp of recurrence relations, mathematical induction, and the behavior of functions. We'll examine a hypothetical sequence and delve into its properties, including how to find its terms, determine convergence, and express it in a closed form (if possible).

Defining the Sequence

Suppose a sequence {a_n} is defined as follows:

• a₁ = x (where x is a real number)
• a_n+1 = f(a_n) for n ≥ 1

This seemingly simple definition opens up a wide range of possibilities, depending on the function f. The key idea here is the recursive nature of the sequence: each term a_n+1 depends directly on the previous term a_n.

Let's unpack this:

• a₁ = x: This is the initial condition or seed of the sequence. It tells us where the sequence begins. The value of x critically influences the behavior of the entire sequence.
• a_n+1 = f(a_n): This is the recurrence relation. It's the rule that dictates how to generate the next term in the sequence, given the current term. f is a function that takes a real number as input (presumably) and returns another real number.

The beauty (and sometimes the difficulty) lies in the choice of the function f. Linear functions, polynomials, trigonometric functions, exponential functions – any function can be used to define a sequence in this way. Different functions lead to radically different sequence behaviors.

Examples to Illustrate

Let's look at a few examples to make this more concrete.

Example 1: A Simple Linear Sequence

Suppose f(y) = 2y + 1, and a₁ = 1. Then:

• a₁ = 1
• a₂ = f(a₁) = f(1) = 2(1) + 1 = 3
• a₃ = f(a₂) = f(3) = 2(3) + 1 = 7
• a₄ = f(a₃) = f(7) = 2(7) + 1 = 15

The sequence is: 1, 3, 7, 15, ... We can see a pattern emerging.

Example 2: A Quadratic Sequence

Suppose f(y) = y², and a₁ = 2. Then:

• a₁ = 2
• a₂ = f(a₁) = f(2) = 2² = 4
• a₃ = f(a₂) = f(4) = 4² = 16
• a₄ = f(a₃) = f(16) = 16² = 256

The sequence is: 2, 4, 16, 256, ... This sequence grows much faster than the linear one.

Example 3: A Trigonometric Sequence

Suppose f(y) = cos(y), and a₁ = 0. Then:

• a₁ = 0
• a₂ = f(a₁) = f(0) = cos(0) = 1
• a₃ = f(a₂) = f(1) = cos(1) ≈ 0.5403
• a₄ = f(a₃) = f(0.5403) = cos(0.5403) ≈ 0.8576

The sequence is: 0, 1, 0.5403, 0.8576, ... This sequence behaves differently again; it oscillates and converges towards a specific value (more on that later).

Analyzing the Sequence

Once we have a sequence defined in this way, several questions arise:

1. Can we find a general formula for a_n? This is called finding a closed-form expression for the sequence. It allows us to calculate a_n directly without having to compute all the preceding terms. Unfortunately, finding a closed-form expression is often very difficult or even impossible for many functions f.
2. Does the sequence converge? In other words, does a_n approach a specific limit as n approaches infinity? If it does, what is the limit?
3. What are the properties of the sequence? Is it monotonic (increasing or decreasing)? Is it bounded? Is it periodic?

Let's address these questions, although a full treatment would require more advanced mathematical tools.

Finding a Closed-Form Expression

For some simple functions f, we can find a closed-form expression. Let's revisit Example 1, where f(y) = 2y + 1 and a₁ = 1. The sequence was: 1, 3, 7, 15, ... We might notice that these numbers are all one less than a power of 2. Specifically:

• a₁ = 2¹ - 1
• a₂ = 2² - 1
• a₃ = 2³ - 1
• a₄ = 2⁴ - 1

This suggests the closed-form expression a_n = 2ⁿ - 1. To rigorously prove this, we can use mathematical induction:

• Base Case: When n = 1, the formula gives a₁ = 2¹ - 1 = 1, which is correct.

• Inductive Hypothesis: Assume that the formula holds for some k ≥ 1; that is, assume a_k = 2^k - 1.

• Inductive Step: We need to show that the formula also holds for n = k + 1. That is, we need to show that a_k+1 = 2^k+1 - 1.

  Using the recurrence relation:

  a_k+1 = f(a_k) = 2a_k + 1

  Substituting the inductive hypothesis:

  a_k+1 = 2(2^k - 1) + 1 = 2^k+1 - 2 + 1 = 2^k+1 - 1

  This is exactly what we wanted to show.

Therefore, by mathematical induction, the closed-form expression a_n = 2ⁿ - 1 is valid for all n ≥ 1.

However, finding a closed-form expression is often much harder. For Example 2 (f(y) = y², a₁ = 2), the sequence grows so rapidly that a closed-form is relatively easy to derive: a_n = 2^{(2^n-1)}. But for more complicated functions like trigonometric functions or mixtures of functions, finding a closed-form expression can be extremely challenging or impossible.

Convergence of the Sequence

Determining whether a sequence converges and, if so, finding its limit is a fundamental question in analysis. A sequence {a_n} converges to a limit L if, for any arbitrarily small positive number ε, there exists a positive integer N such that |a_n - L| < ε for all n > N. Intuitively, this means that as n gets larger and larger, the terms of the sequence get closer and closer to L.

Let's consider the case where the sequence does converge to a limit L. That is, lim_n→∞ a_n = L. Then, since a_n+1 = f(a_n), and assuming f is continuous, we have:

lim_n→∞ a_n+1 = lim_n→∞ f(a_n)

L = f(lim_n→∞ a_n)

L = f(L)

This is a crucial result! If the sequence converges, the limit L must be a fixed point of the function f. A fixed point of a function f is a value L such that f(L) = L.

Important Note: Just because L is a fixed point of f does not guarantee that the sequence will converge to L. It only says that if the sequence converges, it must converge to a fixed point.

Example: Convergence with f(y) = cos(y)

Recall Example 3, where f(y) = cos(y). The equation L = cos(L) has a unique solution (approximately L ≈ 0.739085). This means that if the sequence defined by a_n+1 = cos(a_n) converges, it must converge to this value. It turns out that, regardless of the initial value a₁, this sequence does converge to this fixed point. This can be shown using the Banach fixed-point theorem, but that's beyond the scope of this discussion.

Example: Divergence with f(y) = 2y + 1

In Example 1, f(y) = 2y + 1. The equation L = 2L + 1 implies L = -1. However, the sequence 1, 3, 7, 15, ... clearly diverges to infinity. The fact that L = -1 is a fixed point doesn't mean the sequence converges to it, because the conditions for convergence are not met. The initial term a₁=1 is not equal to -1, therefore the sequence diverges.

Properties of the Sequence

Beyond convergence, we can investigate other properties of the sequence:

• Monotonicity: A sequence is monotonic if it is either always increasing or always decreasing. A sequence {a_n} is increasing if a_n+1 ≥ a_n for all n, and decreasing if a_n+1 ≤ a_n for all n.
• Boundedness: A sequence is bounded if there exist real numbers M and N such that M ≤ a_n ≤ N for all n. In other words, the terms of the sequence are all contained within a finite interval.
• Periodicity: A sequence is periodic if there exists a positive integer p such that a_n+p = a_n for all n. The smallest such p is called the period of the sequence.

Determining these properties often requires careful analysis of the function f and the initial value a₁.

Example: Monotonicity and Boundedness

Consider f(y) = y/2 and a₁ = 1. The sequence is 1, 1/2, 1/4, 1/8, ... This sequence is:

• Decreasing (monotonic) because each term is smaller than the previous term.
• Bounded because 0 ≤ a_n ≤ 1 for all n.
• Convergent (it converges to 0).

Example: Periodicity

Consider f(y) = 1 - y and a₁ = 0. The sequence is 0, 1, 0, 1, ... This sequence is:

• Not monotonic.
• Bounded because 0 ≤ a_n ≤ 1 for all n.
• Periodic with period 2.
• Not convergent (it oscillates between 0 and 1).

The Importance of the Function f

The function f is the heart of the sequence definition. Its properties heavily influence the sequence's behavior.

• Linear Functions: Linear functions f(y) = my + b lead to sequences that either grow exponentially, decay exponentially, or converge to a fixed point (depending on the value of m).
• Polynomial Functions: Polynomial functions can generate sequences with more complex growth patterns. Quadratic functions, for example, can lead to sequences that grow very rapidly.
• Trigonometric Functions: Trigonometric functions introduce oscillatory behavior. Sequences defined using trigonometric functions often converge to fixed points or exhibit periodic behavior.
• Rational Functions: Rational functions (ratios of polynomials) can lead to sequences with interesting asymptotic behavior.
• Piecewise Functions: Piecewise functions can create sequences with unpredictable behavior that changes depending on the range of values.

Applications

Sequences defined by recurrence relations have numerous applications in various fields:

• Computer Science: They are used in algorithms, data structures (e.g., linked lists), and generating random numbers. The Fibonacci sequence, a classic example, is used in search algorithms and data compression.
• Physics: They appear in modeling physical systems, such as the motion of a pendulum or the population growth of a species.
• Economics: They are used to model economic growth, inflation, and other economic phenomena.
• Mathematics: They are fundamental to calculus, real analysis, and dynamical systems. Iterative methods for solving equations (like Newton's method) are based on generating sequences that converge to the solution.
• Biology: Population modeling is heavily reliant on recursive sequences.

Common Pitfalls

Analyzing sequences defined by recurrence relations can be tricky. Here are some common pitfalls to avoid:

• Assuming Convergence: Don't assume a sequence converges just because it seems to be approaching a value. You need to rigorously prove convergence using mathematical tools.
• Ignoring Initial Conditions: The initial condition a₁ is crucial. Changing a₁ can drastically alter the sequence's behavior.
• Incorrectly Applying Fixed-Point Theorem: Remember that a fixed point is a necessary but not sufficient condition for convergence.
• Difficulty Finding Closed-Form Expressions: Don't get discouraged if you can't find a closed-form expression. Many sequences don't have one. Focus on analyzing the sequence's behavior using other techniques.
• Overlooking Oscillatory Behavior: Some sequences oscillate without converging. Make sure to check for this possibility.

Strategies for Analysis

Here are some strategies for analyzing sequences defined as a_n+1 = f(a_n):

1. Calculate the First Few Terms: This will give you a feel for the sequence's behavior.
2. Graph the Function f(y): Visualizing the function can help you understand how the sequence evolves. Plot y = f(x) and the line y = x. The intersection points are the fixed points.
3. Find Fixed Points: Solve the equation L = f(L).
4. Analyze the Derivative of f(y): The derivative f'(y) at the fixed points provides information about the stability of the fixed points. If |f'(L)| < 1, the fixed point is likely to be attracting. If |f'(L)| > 1, the fixed point is likely to be repelling.
5. Try to Prove Monotonicity and Boundedness: If you can show that the sequence is monotonic and bounded, then it must converge (by the Monotone Convergence Theorem).
6. Use Mathematical Induction: If you suspect a closed-form expression, use mathematical induction to prove it.
7. Look for Patterns: Sometimes, you can spot a pattern in the sequence that helps you understand its behavior.
8. Use Numerical Methods: If analytical methods fail, use numerical methods (e.g., a computer program) to approximate the terms of the sequence and investigate its behavior.

FAQ

Q: What is a recurrence relation?

A: A recurrence relation is an equation that defines a sequence recursively. Each term of the sequence is defined in terms of previous terms.

Q: What is a closed-form expression?

A: A closed-form expression is a formula that allows you to calculate the n-th term of a sequence directly, without having to compute all the preceding terms.

Q: How do I determine if a sequence converges?

A: Several methods can be used. Some common techniques include:

• Showing that the sequence is monotonic and bounded (Monotone Convergence Theorem).
• Using the Squeeze Theorem.
• Finding the limit using algebraic manipulations.
• Analyzing the derivative of the function f at the fixed points.

Q: What is a fixed point?

A: A fixed point of a function f is a value L such that f(L) = L. If a sequence defined by a_n+1 = f(a_n) converges, its limit must be a fixed point of f.

Q: Can all sequences defined by recurrence relations be expressed in closed form?

A: No, many sequences do not have a closed-form expression.

Q: Is it always possible to determine if a sequence converges or diverges?

A: No. Some sequences are very difficult to analyze, and it may not be possible to determine definitively whether they converge or diverge.

Conclusion

Defining a sequence as a_n+1 = f(a_n) is a powerful and versatile technique. By carefully choosing the function f and the initial value a₁, we can generate a wide variety of sequences with different properties. Analyzing these sequences involves finding closed-form expressions (if possible), determining convergence, and investigating properties like monotonicity, boundedness, and periodicity. Understanding these concepts is crucial in various fields, including mathematics, computer science, physics, and economics. While analyzing such sequences can be challenging, the strategies outlined in this discussion provide a framework for understanding their behavior and uncovering their hidden properties. The key is to be patient, persistent, and to leverage a combination of analytical and numerical methods.