The mathematical constant φ (phi), the golden ratio, has captivated thinkers for millennia. Now, a new analysis reveals the exact algorithmic behavior of a classic iterative sequence converging to φ. The sequence, defined by $x_0 = 1$ and $x_{n+1} = 1 + \frac{1}{x_n}$ for $n > 0$, approaches φ with remarkable predictability.

"The ratio of the error at one step to the error at the previous step is approximately 0.309, meaning the error shrinks by nearly 70% each iteration."

This precision comes from Taylor series expansion applied to the error term. When the approximation $x_n = φ + ε$ is substituted into the recurrence relation, the error propagation factor emerges as:

$$\frac{ε_{n+1}}{ε_n} \approx \frac{\sqrt{5}-3}{2} \approx -0.309$$

The negative sign indicates oscillatory convergence, while the magnitude reveals the exponential decay rate. Higher-order terms in the expansion (approximately $-0.03ε$) cause slight acceleration in early iterations, but the dominant 0.309 factor quickly establishes stable convergence behavior.

This analysis, while purely mathematical, has implications for computational mathematics and numerical stability in algorithms involving recursive sequences. Understanding convergence rates is crucial for developing efficient approximation methods in computational geometry, optimization algorithms, and even certain cryptographic protocols where irrational approximations are used.

The golden ratio's emergence in this iterative process—appearing in nature's growth patterns, art, and architecture—now gains a new layer of algorithmic transparency. Researchers note that similar convergence analysis could be applied to other quadratic irrationals defined by continued fraction expansions.