Problem 100 - Thomas Calculus

Question

A sequence of rational numbers is described as follows: $$\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \ldots, \frac{a}{b}, \frac{a+2 b}{a+b}, \ldots$$ Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let \(x_{n}\) and \(y_{n}\) be, respectively, the numerator and the denominator of the \(n\) th fraction \(r_{n}=x_{n} / y_{n}\) a. Verify that \(x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1\) and, more generally, that if \(a^{2}-2 b^{2}=-1\) or \(+1,\) then $$(a+2 b)^{2}-2(a+b)^{2}=+1 \quad\( or \)\quad-1$$ respectively. b. The fractions \(r_{n}=x_{n} / y_{n}\) approach a limit as \(n\) increases. What is that limit? (Hint: Use part (a) to show that \(r_{n}^{2}-2=\pm\left(1 / y_{n}\right)^{2}\) and that \(y_{n}\) is not less than \(n .\) )

Step-by-Step Solution

Answer

The limit of the sequence as \( n \rightarrow \infty \) is \( \sqrt{2} \).

Step 1: Verify Initial Conditions

For the first fraction, \( \frac{1}{1} \), we have \( x_1 = 1 \) and \( y_1 = 1 \). Calculate \( x_1^2 - 2y_1^2 = 1^2 - 2(1)^2 = 1 - 2 = -1 \). This matches the given initial condition for \( x_1^2 - 2y_1^2 = -1 \).For the second fraction, \( \frac{3}{2} \), we have \( x_2 = 3 \) and \( y_2 = 2 \). Calculate \( x_2^2 - 2y_2^2 = 3^2 - 2(2)^2 = 9 - 8 = 1 \). This matches the second condition for \( x_2^2 - 2y_2^2 = +1 \).

Step 2: General Case Verification

Assume \( a^2 - 2b^2 = -1 \) or \( +1 \). The expression for the next terms in the sequence is \( x_{n+1} = a + 2b \) and \( y_{n+1} = a + b \). We need to verify \((a + 2b)^2 - 2(a + b)^2 = \pm1\).First, expand both squares:\( (a + 2b)^2 = a^2 + 4ab + 4b^2 \)\( 2(a + b)^2 = 2(a^2 + 2ab + b^2) = 2a^2 + 4ab + 2b^2 \)Subtract the second expression from the first:\( (a^2 + 4ab + 4b^2) - (2a^2 + 4ab + 2b^2) = a^2 + 4ab + 4b^2 - 2a^2 - 4ab - 2b^2 \)\( = -a^2 + 2b^2 = -(a^2 - 2b^2) \)Thus, if \( a^2 - 2b^2 = -1 \), \( -(a^2 - 2b^2) = 1 \), and if \( a^2 - 2b^2 = 1 \), then \( -(a^2 - 2b^2) = -1 \). The proposition holds.

Step 3: Limit as Sequence Progresses

To find the limit as \( n \rightarrow \infty \), consider the expression \( r_n^2 - 2 = \pm \left( \frac{1}{y_n} \right)^2 \).As \( y_n \) increases, the term \( \left( \frac{1}{y_n} \right)^2 \) approaches 0. Therefore, \( r_n^2 - 2 = 0 \), giving \( r_n^2 = 2 \), or \( r_n = \sqrt{2} \) since \( r_n > 0 \).The hint tells us that \( y_n \geq n \), confirming that \( y_n \rightarrow \infty \). Therefore, the limit of \( r_n \) as \( n \rightarrow \infty \) is \( \sqrt{2} \).

Key Concepts

Rational Numbers

Rational numbers are part of the number system and can be expressed as a fraction of two integers, where the numerator is divided by a non-zero denominator. For example, numbers like \( \frac{1}{2} \), \( \frac{-5}{3} \), and \( \frac{7}{1} \) are all rational numbers. These numbers are crucial in sequences, especially when analyzing patterns and progressions in mathematics. Rational numbers:

• Aren't limited to positive values; they include negative values and zero.
• Can be represented as either terminating or repeating decimals, demonstrating their precise nature.
• Are vital in defining arithmetic series where each term in the sequence is related by either addition or subtraction.

In sequences of rational numbers, the study often revolves around how these numbers interact and form patterns, as seen in the given sequence.

Limits of Sequences

The limit of a sequence is a fundamental concept in calculus and mathematical analysis. It describes the behavior of a sequence as it approaches a specific value as the index number increases indefinitely. In the context of rational sequences, the limit helps identify what value the sequence is tending towards. Consider:

• The limit of the sequence of fractions \( r_n = \frac{x_n}{y_n} \) as \( n \to \infty \).
• This sequence approaches \( \sqrt{2} \) because the deviation \( r_n^2 - 2 = \pm \left( \frac{1}{y_n} \right)^2 \) becomes negligible as \( y_n \) grows larger, ultimately turning the equation into \( r_n^2 = 2 \).
• The behavior of the sequence is heavily influenced by the growth rate of the denominator \( y_n \), which, according to the exercise, is not less than \( n \), ensuring \( y_n \to \infty \).

Hence, the limit helps connect the sequence's trend to a real-world value or mathematical constant, invaluable for deeper understanding and analysis of sequence behavior.

Recursive Sequences

Recursive sequences are sequences where each term is defined as a function of its preceding terms. Essentially, the sequence is generated by repeatedly applying a specific rule or formula. This exercise illustrates the idea of recursion through:

• Defining \( x_{n+1} = a + 2b \) and \( y_{n+1} = a + b \), i.e., each subsequent term depends on the previous ones.
• The expressions show how the sequences of numerators and denominators progress according to their defined rules.
• Recursive formulas require initial conditions to initiate the sequence, such as the first term or two, from which all other terms are systematically derived.

Recursive sequences not only offer a way to predict the next elements but also help establish relationships between terms, aiding in the verification of properties, such as checking whether the derived equation \( (a+2b)^2 - 2(a+b)^2 = \pm 1 \) holds true.