Putnam 1995, Problem B-4

Evaluate
\[
\sqrt[8]{2207 - \frac{1}{2207 - \frac{1}{2207 - \cdots}}}.
\]
Express your answer in the form \( \frac{a + b\sqrt{c}}{d} \), where \( a,b,c,d \) are integers.

Solution:
The infinite continued fraction is defined as the limit of the sequence \( L_0 = 2207, L_{n+1} = 2207 - 1/L_n \). Notice that the sequence is strictly decreasing (by induction) and thus indeed has a limit \( L \), which satisfies \( L = 2207 - 1/L \), or rewriting, \( L^2 - 2207L + 1 = 0 \). Moreover, we want the greater of the two roots.