Tea Time Numerical Analysis · Problem 1.1.20

Preliminaries — Leon Brin

Mathematics — Numerical Analysis · relative-error

Problem

The golden ratio,

\dfrac{1 + \sqrt{5}}{2}

, is found in nature and in mathematics in a variety of places. For example, if

F_n

is the

n^{th}

Fibonacci number, then

\lim_{n\to\infty} \frac{F_{n+1}}{F_n} = \frac{1+\sqrt{5}}{2}

Therefore,

F_{11}/F_{10}

may be used as an approximation of the golden ratio. Find the relative error in this approximation. HINT: The Fibonacci sequence is defined by

F_0 = 1

F_1 = 1

F_n = F_{n-1} + F_{n-2}

for

n \geq 2