Continued Fraction Comparisons

★ 6/10number theory Monotonicity sequences Continued Fractions

Problem

Define a sequence $\{b_n\}$ of continued fractions by

b_1 = 1 + \frac{1}{\alpha_1}, \quad b_2 = 1 + \frac{1}{\alpha_1 + \frac{1}{\alpha_2}}, \quad b_3 = 1 + \frac{1}{\alpha_1 + \frac{1}{\alpha_2 + \frac{1}{\alpha_3}}}, \quad \dots

where each $\alpha_k \in \mathbb{N}^*$ . Determine, with proof, which of the following always hold:

1. $b_1 < b_5$ ; 2. $b_3 < b_8$ ; 3. $b_6 < b_2$ ; 4. $b_4 < b_7$ .

Answer

Solution

Difficulty	6/10
Topics	number theory, Monotonicity, sequences, Continued Fractions

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