A sequence of rational numbers is described as follows:
1/1,3/2,7/5,17/12,…,a/b,(a + 2b)/(a + b),…
Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xₙ and yₙ be, respectively, the numerator and the denominator of the nᵗʰ fraction rₙ = xₙ / yₙ.
b. The fractions rₙ = xₙ / yₙ approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that rₙ² − 2 = ±(1 / yₙ)² and that yₙ is not less than n.)

Verified step by step guidance

1. First, recognize that the sequence of fractions is defined recursively by the relation: \(r_{n+1} = \frac{a + 2b}{a + b}\), where \(r_n = \frac{x_n}{y_n} = \frac{a}{b}\). This means \(r_{n+1} = \frac{x_n + 2 y_n}{x_n + y_n}\).
2. Express \(r_{n+1}\) in terms of \(r_n\) by dividing numerator and denominator by \(y_n\): \(r_{n+1} = \frac{\frac{x_n}{y_n} + 2}{\frac{x_n}{y_n} + 1} = \frac{r_n + 2}{r_n + 1}\). This gives a recursive formula for \(r_n\).
3. To find the limit \(L\) of \(r_n\) as \(n\) approaches infinity, assume the limit exists and satisfies \(L = \frac{L + 2}{L + 1}\). Solve this equation for \(L\) by multiplying both sides by \((L + 1)\) to get \(L(L + 1) = L + 2\).
4. Simplify the equation: \(L^2 + L = L + 2\), which reduces to \(L^2 = 2\). Therefore, the possible limits are \(L = \sqrt{2}\) or \(L = -\sqrt{2}\). Since the sequence terms are positive, the limit is \(L = \sqrt{2}\).
5. Use the hint from part (a) that \(r_n^2 - 2 = \pm \frac{1}{y_n^2}\) and that \(y_n\) grows at least as fast as \(n\), so \(\frac{1}{y_n^2} \to 0\) as \(n \to \infty\). This confirms that \(r_n^2\) approaches 2, reinforcing that \(r_n\) approaches \(\sqrt{2}\).

Key Concepts

Recursive Sequences and Their Limits

A recursive sequence is defined by a relation that expresses each term using previous terms. Understanding how to analyze such sequences helps determine their behavior as n grows large, including whether they converge to a limit. In this problem, both numerators and denominators follow recursive patterns, which influence the limit of their ratio.

Limits of Sequences and Convergence

The limit of a sequence is the value that its terms approach as the index n tends to infinity. To find the limit of the fraction sequence rₙ = xₙ / yₙ, one must analyze how the numerator and denominator grow and use given relations to show convergence. Recognizing that yₙ grows at least linearly helps in bounding terms and proving the limit exists.

Using Algebraic Identities to Find Limits

Algebraic identities, such as the given relation rₙ² − 2 = ±(1 / yₙ)², are crucial for linking sequence terms and simplifying expressions. This identity connects the ratio rₙ to its denominator yₙ, allowing us to approximate rₙ² and deduce the limit by considering the behavior of 1 / yₙ² as n increases.