Limit of a Recursive Sequence

Given the recursive sequence defined by $a_1 = 1$ and $a_n = 1 + \frac{1}{1 + a_{n-1}}$ for $n \geq 2$, determine the limit of the sequence as $n \to \infty$.

This problem invites you to explore the characteristics and behavior of a recursively defined sequence. The focus is on understanding sequences and their limits, which is a cornerstone concept in real analysis. Recursive sequences are defined such that future terms depend on preceding values, which provides a good way to cultivate understanding of dynamic progression and accumulation in sequences. To solve the problem, one must grasp the concept of convergence of sequences and apply the limit laws.

In this context, achieving the limit of an infinity-proceeding sequence requires thorough exploration and manipulation of its recursive definition. A visualization strategy can be quite effective—to identify a prospective limit, suppose that the sequence indeed approaches some limit L as $n$ approaches infinity, leading to the equation $L = 1 + \frac{1}{1 + L}$. By solving for L, it is possible to get insights into sequence behavior although showing such a limit exists may demand additional justification, perhaps involving bounds and monotonicity.

This not only helps in determining the value of L but also solidifies understanding of the convergence process itself. This type of problem illuminates key analytical mechanics in recursive structures and is a perfect showcase for convergence concepts in sequence analysis.