When we explore recursive sequences, we discover some fascinating patterns about how they behave.

So, what are recursive sequences?

At their core, they start with a base case—a starting point—and a rule that helps us find the next numbers based on the ones before. This leads to interesting results about whether the sequence settles down to a certain value or keeps changing.

Let's look at the well-known Fibonacci sequence. It works like this:

• The first number, F(0), is 0.
• The second number, F(1), is 1.
• For any number after that, F(n) = F(n-1) + F(n-2) (for n 2 or bigger).

In simpler terms, each number is the sum of the two numbers right before it. As we go further along this sequence, we notice that even though Fibonacci numbers get really big, the ratio between them gets closer to a specific value called the golden ratio, which is about 1.618. This means some recursive sequences can get closer to certain values over time.

However, not all recursive sequences are easy like the Fibonacci ones. For example, let's look at another sequence defined as follows:

• Start with a_0 = 1.
• Then use the rule a_n = 2a_(n-1) + 1 to find the next numbers.

Here's how it goes:

• a_1 = 2(1) + 1 = 3
• a_2 = 2(3) + 1 = 7
• a_3 = 2(7) + 1 = 15.

From this pattern, we see that as we keep increasing n, a_n just keeps getting bigger and bigger, heading towards infinity. This shows us that different rules can lead to different growth patterns.

When we talk about convergence (when sequences settle down to a limit), we can use tools like the Squeeze Theorem and Fixed Point Theorems. A sequence is said to converge to a limit L if, no matter how small a distance we choose (called epsilon), there is a point N in the sequence where all terms after that point get closer to L. In terms of recursive definitions, if our rule lets the sequence stabilize around a number, it will converge.

Another important idea is the contraction mapping principle. It tells us that if the function we’re using in the recursion brings numbers closer together each time, then we will probably see convergence happen. This principle helps us understand how stable various recursive sequences can be, especially when solving math problems.

Now, let's talk about divergence, which means the sequence doesn't settle. It can happen in two main ways:

1. Oscillation: Some sequences jump around without settling down. For example, with the sequence a_n = (-1)^n, it keeps switching between 1 and -1, so there’s no clear limit.

2. Unbounded Growth: Like the earlier example where the sequence grows forever, we might find sequences that just keep getting larger and larger. Another instance is b_n = 3b_(n-1) + 1, which also increases dramatically.

To understand how recursive sequences work, it's key to identify their growth patterns and understand whether they can stabilize or not. As you look through different examples, it’s important to think about their behaviors, since this will help you learn more.

The ideas about convergence and divergence from recursive sequences go far beyond just math. They help us think critically in areas like computer algorithms, economic models, and other systems that change over time. Studying recursion not only deepens our math skills but also shows us how math is a vibrant and evolving field.