Fibonacci Ratio Convergence

Calculus 2 · Axiom Academy

EXAMPLE Fibonacci Ratio Convergence Discovering how consecutive Fibonacci ratios approach the golden ratio The Fibonacci sequence is defined by the recurrence F_ n+1 = F_n + F_ n-1 . Show that the ratio of consecutive terms F_ n+1 /F_n converges, and find its limit. Each term is the sum of the two preceding terms Excellent work! You've discovered one of mathematics' most beautiful connections. Here's what we learned: Convergence pattern: the ratio F_ n+1 /F_n converges to the golden ratio as , demonstrating limit behavior in a recursive sequence. Algebraic derivation: assuming the limit exists and applying it to the recurrence relation gives , the equation that defines the golden ratio. The golden ratio: appears throughout mathematics, art, and nature thanks to its unique property . Rate of convergence: the ratios approach quickly — is already accurate to several decimal places. This connection between Fibonacci numbers and the golden ratio illustrates how discrete sequences can have continuous limits — a fundamental idea in calculus and analysis. In a precise sense, is the "most irrational" number of all.

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